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<p>Some provinces have names with First Nations origins. For example, &quot;Ontario&quot; comes from an Iroquois word meaning “beautiful water&quot; or &quot;beautiful lake.&quot; If the number of provincial names with First Nations origins is <code class='latex inline'> a </code> , and the number with other origins is <code class='latex inline'> b </code> , the numbers are related by the following equations.</p><p><code class='latex inline'>\displaystyle a+b=10\\3 a-2 b=0 </code></p><p>a) Communication Interpret each equation in words.</p><p>b) Find the number of provinces that have names with First Nations origins.</p>
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<p>A speedboat took 3 h to travel a distance of 60 km up a river, against the current. The return trip took 2 h. Find the average speed of the boat in still water and the speed of the current.</p>
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<p>These equations represent the total cost to charter buses from three different bus companies. In each equation, <code class='latex inline'>\displaystyle C </code> represents the total cost in dollars, and <code class='latex inline'>\displaystyle n </code> represents the number of passengers.</p><p>Company X: <code class='latex inline'>\displaystyle C=35 n+500 </code> </p><p>Company Y: <code class='latex inline'>\displaystyle C=25 n+2000 </code> </p><p>Company Z: <code class='latex inline'>\displaystyle C=37.50 n </code></p><p>Layton spends <code class='latex inline'>\displaystyle \$10150 </code> to charter buses to transport</p><p>326 passengers to a reunion.</p><p>a) Which bus company did Layton use?</p><p>b) Did Layton choose the best company if he wanted to spend as little as possible to charter the buses? Explain your answer.</p> Similar Questions Learning Path L1 Quick Intro to Factoring Trinomial with Leading a L2 Introduction to Factoring ax^2+bx+c L3 Factoring ax^2+bx+c, ex1 Now You Try <p>A speedboat took 3 h to travel a distance of 60 km up a river, against the current. The return trip took 2 h. Find the average speed of the boat in still water and the speed of the current.</p> <p>A grocery bag contains five apples, one pineapple, and one orange. The total mass of the fruit is 2.7 kg. The mass of the pineapple is 1.25 kg, and the mass of the orange is 0.2 kg. Create and solve an equation to determine the average mass of each apple.</p> <p>A fitness centre charges an initiation fee of plus a monthly charge of <code class='latex inline'> \$ y </code> . Write algebraic expressions to show how much it costs to be a member for</p><p>b) 15 months</p>
<p>The two largest deserts in the world are the Sahara Desert and the Australian Desert. The sum of their areas is 13 million square kilometres. The area of the Sahara Desert is 5 million square kilometres more than the area of the Australian Desert. Solve the following system graphically to find the area of each desert, in millions of square kilometres.</p><p><code class='latex inline'>\displaystyle s+a=13\\s=a+5 </code></p>
<p>A recording artist is offered two deals for her fourth CD release:</p> <ul> <li>Royalty only: <code class='latex inline'>\$1</code> per CD sold</li> <li><p>Partial royalty: <code class='latex inline'>\$2000</code> plus <code class='latex inline'>\$0.50</code> per CD sold</p></li> <li><p>Find the solution to the linear system and explain what it means.</p></li> </ul> <img src="/qimages/59836" /> <ol> <li>Measurement <code class='latex inline'> \angle x </code> and <code class='latex inline'> \angle y </code> are two acute angles in a right triangle. The measures of the angles are related by the following system of equations.</li> </ol> <p><code class='latex inline'>\displaystyle \begin{array}{l} x+y=90 \\ y-6=3 x \end{array} </code></p><img src="/qimages/59837" /><p>a) Communication Interpret each equation in words.</p><p>b) Solve the system of equations to find the measure of each acute angle.</p> <p>Ms. Frizzle has three daughters: Allison, Belle, and Claire. Today, January 1, their ages are, respectively,</p><p><code class='latex inline'> \displaystyle \begin{array}{cccccc} & A(n) = -(n + 30)+(2n + 5) \\ & B(n) = (7 -n)-(32 -2n) \\ & C(n) = (n - 26)-(n + 4) + (n - 3) \\ \end{array} </code></p><p>All ages are expressed in years, and n represents Ms. Frizzle&#39;s age.</p><p>a) Are the daughter triples? Explain.</p><p>b) Are any of them twins? Explain.</p> <p>Zidane works in a sporting goods store selling skates. A pair of hockey skates costs <code class='latex inline'> \$ 58.00 </code> and a pair of figures skates costs <code class='latex inline'> \$56.00 </code> . One shift, Zidane sold 32 pairs of skates. His receipts totalled <code class='latex inline'> \$ 1828 </code> , not including taxes.</p><p>a) How many pairs of hockey skates did Zidane sell?</p><p>b) How many pairs of figure skates did Zidane sell?</p>
<p>On weekends, as part of his exercise routine, Carl goes for a run, partly El on paved trails and partly across rough terrain. He runs at 10 km/h on the trails, but his speed is reduced to 5 km/h on the rough terrain. One day, he ran 12 km in 1.5 h. How far did he run on the rough terrain? </p>
<p>International basketball competitions are played on a rectangular court where the length is 2m less than twice the width.</p><p>a) If the perimeter of the court is 867m, what are the dimensions of the courts?</p><p>b) Solve this problem using a different method.</p><p>c) Compare the methods. Describe one advantage and one disadvantage of each approach.</p>
<p>Translation each sentence into an equation.</p><p>Tell how you are assigning the two variables.</p><p>a) The perimeter of a basketball court is <code class='latex inline'>40</code> m.</p><p>b) The average of two numbers is <code class='latex inline'>15</code>.</p><p>c) The value of the quarters and loonies in a vending machine is <code class='latex inline'>\$37</code>.</p><p>d) The total receipts from adult tickets at <code class='latex inline'>\$20</code> each and student tickets at <code class='latex inline'>\$12</code> each was <code class='latex inline'>\$9250</code>.</p>
<p>A magic square is an array of numbers with the same sum across any row, column, or main diagonal. </p><img src="/qimages/701" /><p><strong>a)</strong> Determine a system of linear equations you can use to determine the values of A and B in both squares. </p><p><strong>b)</strong> What are the values of <code class='latex inline'>A</code> and <code class='latex inline'>B</code>?</p>
<p>Kristen has a total of $1000 to invest. She puts part of it in an account paying 4% interest/year and the rest in a bond paying 6.5% interest. If she has$50 in simple interest at the end of the year, how much was invested at each rate?</p>
<p>The sum of two integers is 42. The difference of the two numbers is 17. </p><p><strong>(a)</strong> Create a system of linear equations to model each statement above. </p><p><strong>(b)</strong> Determine the integers using a graph.</p>
<p>Phoenix Health Club charges a <code class='latex inline'> \$200 </code> initiation fee, plus <code class='latex inline'> \$ 15 </code> a month. Champion Health Club charges a \$100 initiation fee, plus <code class='latex inline'> \$ 20 </code> a month. The costs can be compared using the following equations.</p><p>Phoenix Cost: <code class='latex inline'> \quad C=200+15 m </code> </p><p>Champion Cost: <code class='latex inline'> C=100+20 \mathrm{~m} </code> </p><p>a) Find the point of intersection of the two lines.</p><p>b) After how many months are the costs the same?</p><p>c) If you joined a club for only a year, which club would be less expensive?</p>
<p>A baseball club pays a vendor <code class='latex inline'>\displaystyle \$50 </code> per game for selling bags of peanuts for <code class='latex inline'>\displaystyle \$ 2.50 </code> each. The club also pays the vendor a commission of <code class='latex inline'>\displaystyle \$0.05 </code> per bag.</p><p>a) Determine a function that describes the income the vendor makes for each baseball game. Define the variables in your function.</p><p>b) Determine a function that describes the revenue the vendor generates each game for the baseball club. Define the variables in your function.</p><p>c) Determine the number of bags of peanuts the vendor must sell before the baseball club makes a profit from his efforts.</p> <p>Mai invests her summer earnings of$2440. She invests part of the money at 8%/year, and the rest at 7.5%/year. After one year, these investments earn $193 in simple interest. How much did she invest at each rate?</p> <ol> <li>Granola One type of granola is <code class='latex inline'>\displaystyle 30 \% </code> fruit, and another type is <code class='latex inline'>\displaystyle 15 \% </code> fruit. What mass of each type of granola should be mixed to make <code class='latex inline'>\displaystyle 600 \mathrm{~g} </code> of granola that is <code class='latex inline'>\displaystyle 21 \% </code> fruit?</li> </ol> <p>A video rental company has two monthly plans:</p> <ul> <li><p>Plan A: <code class='latex inline'>\$40</code> for unlimited rentals</p></li> <li><p>Plan B: <code class='latex inline'>\displaystyle \$10 </code> plus <code class='latex inline'>\displaystyle \$ 3 </code> per video</p></li> </ul> <p>a) Graph this linear system and find the solution.</p><p>b) Explain the conditions under which each plan is better.</p>
<p>The sport of fencing has three main forms: sabre, foil, and épée. Sabre bouts take place within a rectangle of perimeter <code class='latex inline'> 52 \mathrm{~m} </code> . Decide whether the perimeter and each of the following pieces of information are sufficient to allow you to find the dimensions of the rectangle. Explain and justify your reasoning. Find the dimensions of the rectangle, where possible.</p><p>d) The perimeter is <code class='latex inline'> 28 \mathrm{~m} </code> less than the perimeter of a basketball court.</p>
<p>Cersei and her brother Tyrion decide to race home. Cersei is a faster runner than Tyrion, so she gives him a head start. Their distance-time graphs are shown.</p><img src="/qimages/1215" /><p>i) For what length of race will each runner win? For what length of race will they tie?</p><p>ii) Explain the significance of the solution of this linear system.</p>
<p>A new amusement park is going to be built near two major highways. On a coordinate grid of the area, with the scale 1 unit represents 1 km, the park is located at P(3, 4). Highway 2 is represented by the equation <code class='latex inline'>y = 2x + 5</code>, and Highway 10 is represented by the equation <code class='latex inline'>y = -0.5x + 2</code>. Determine the coordinates of the exits that must be built on each highway to result in the shortest road to the park.</p>
<p>Cersei and her brother Tyrion decide to race home. Cersei is a faster runner than Tyrion, so she gives him a head start. Their distance-time graphs are shown.</p><img src="/qimages/1215" /> <ul> <li>How fast does Cersei run?</li> </ul>
<ol> <li>Flying speeds A plane flew <code class='latex inline'>\displaystyle 3000 \mathrm{~km} </code> from Calgary to Montréal with the wind in <code class='latex inline'>\displaystyle 5 \mathrm{~h} </code>. The return flight into the wind took <code class='latex inline'>\displaystyle 6 \mathrm{~h} </code>. Find the wind speed and the speed of the plane in still air.</li> </ol>
<p>The sum of two numbers is <code class='latex inline'> 255 . </code> When the smaller is subtracted from the larger, the result is <code class='latex inline'> 39 . </code> Find the numbers.</p>
<p>Maddy would like to rent a digital camera this weekend. One company charges a flat rate of $75/day. A second company charges$35/day plus $5/h.</p><p>a) Write two equations to represent the information.</p><p>b) Solve the linear system to find the number of hours for which the cost of renting a digital camera is the same for both companies.</p> <p>Find the values of <code class='latex inline'> x </code> and <code class='latex inline'> y </code> .</p><img src="/qimages/62309" /> <p>Patrick owns an apartment building. He knows that the money he earns in a month depends on the rent he charges. This relationship can be modelled by <code class='latex inline'>E=\displaystyle{\frac{1}{50}}R(1650-R)</code>, where <code class='latex inline'>E</code> is Patrick&#39;s monthly earnings, in dollars, and <code class='latex inline'>R</code> is the amount of rent, in dollars, he charges each tenant.</p> <ul> <li>How much will he earn if he sets the rent at <code class='latex inline'>\$900</code>?</li> </ul>
<p>Jenna and Maya have walkie-talkies with a range of <code class='latex inline'>\displaystyle 5 \mathrm{~km} </code>. They leave the park on their bicycles, at the same time. Jenna rides east at <code class='latex inline'>\displaystyle 14 \mathrm{~km} / \mathrm{h} </code> and Maya rides west at <code class='latex inline'>\displaystyle 12 \mathrm{~km} / \mathrm{h} </code>. After half an hour, will they be able to use their walkie-talkies? How do you know?</p>
<p>One type of fertilizer has 40% nitrogen and the second type of fertilizer has 20% nitrogen. How much of each type of fertilizer should be mixed to make 800 kg of fertilizer that has 25% nitrogen?</p>
<p>An air trafi‘ic controller is plotting the course of two jets scheduled to land in 15 min. One aircraft is following a path defined by the equation <code class='latex inline'>3x - 5y = 20</code> and the other by the equation <code class='latex inline'>18x = 30y + 72</code>. Should the controller alter the paths of either aircraft? Justify your decision.</p>
<ol> <li>Longest rivers The Mackenzie, the longest river in Canada, is <code class='latex inline'>\displaystyle 1056 \mathrm{~km} </code> longer than the Yukon, the second longest river. The total length of the two rivers is <code class='latex inline'>\displaystyle 7426 \mathrm{~km} . </code> Find the length of each river.</li> </ol>
<p>Suppose the Central Perk coffee shop sells a cup of espresso for $2.00 and a cup of cappuccino for$2.50. On Friday, Destiny sold 30 more cups of cappuccino than espresso for a total of $178.50 worth of espresso and cappuccino. How many cups of each were sold?</p> <p>A company packs raisins in 500 g cartons and 750 g bags. One day, 887.5 kg of raisins were packed into full cartons and bags. Ben wondered how many cartons and how many bags could have been packed. State the linear equation which represents this situation.</p> <p>The highest point in British Columbia is on Fairweather Mountain, <code class='latex inline'> f </code> metres above sea level. The highest point in Ontario is on Ishpatina Ridge, <code class='latex inline'> i </code> metres above sea level. The relationship between the heights can be modelled by the following system of equations.</p><p><code class='latex inline'>\displaystyle \begin{array}{l} f-i=3970 \\ f=7 i-188 \end{array} </code></p> <p>The rectangle has an area of <code class='latex inline'> m </code> square units and a perimeter of <code class='latex inline'> 2 m </code> units. What is the value of <code class='latex inline'> x </code> ?</p><img src="/qimages/62318" /> <p>At Lisa&#39;s Sub Shop, two ham subs and four roast-beef subs cost <code class='latex inline'> \$ 34 </code> . Five ham subs and 6 roast-beef subs cost <code class='latex inline'> \$61 </code> . If one ham sub costs <code class='latex inline'> \$ x </code> and one roast-beef sub costs <code class='latex inline'> \$y </code> , the information can be modelled by the following system of equations.</p><p><code class='latex inline'>\displaystyle 2 x+4 y=34\\5 x+6 y=61 </code></p><p>Solve the system of equations to find the cost of each type of sub.</p> <p>The hourly rate of a waiter is <code class='latex inline'>\displaystyle \$ 4 </code> plus tips. On a particular day, the waiter worked 8 hours and received more than <code class='latex inline'>\displaystyle \$150 </code> in pay. Which could be the amount of tips the waiter received?</p><p><code class='latex inline'>\displaystyle \begin{array}{llll}\text { (A) } \$ 18.75 & \text { (B) } \$32 & \text { (C) } \$ 118 & \text { D } \$120.75\end{array} </code></p> <ol> <li>Lawn fertilizer One lawn fertilizer is <code class='latex inline'>\displaystyle 24 \% </code> nitrogen, and another is <code class='latex inline'>\displaystyle 12 \% </code> nitrogen. How much of each fertilizer should be mixed to obtain <code class='latex inline'>\displaystyle 100 \mathrm{~kg} </code> of fertilizer that is <code class='latex inline'>\displaystyle 21 \% </code> nitrogen?</li> </ol> <p>Jon downloads music to his MP3 player from a site that charges$12.95 per month and $0.45 per song. Another site charges$8.99 per month and $0.95 per song. Compare the cost of the two sites using a table and a graph.</p> <p>Graham and Colin leave the same place at the same time and drive in opposite directions. Colin drives <code class='latex inline'>\displaystyle 10 \mathrm{~km} / \mathrm{h} </code> faster than Graham does. After <code class='latex inline'>\displaystyle 2 \mathrm{~h} </code>, they are <code class='latex inline'>\displaystyle 200 \mathrm{~km} </code> apart. How fast is each man driving?</p> <p>The sport of fencing has three main forms: sabre, foil, and épée. Sabre bouts take place within a rectangle of perimeter <code class='latex inline'> 52 \mathrm{~m} </code> . Decide whether the perimeter and each of the following pieces of information are sufficient to allow you to find the dimensions of the rectangle. Explain and justify your reasoning. Find the dimensions of the rectangle, where possible.</p><p>b) The sum of the length and the width is <code class='latex inline'> 26 \mathrm{~m} </code> .</p> <p>Cersei and her brother Tyrion decide to race home. Cersei is a faster runner than Tyrion, so she gives him a head start. Their distance-time graphs are shown.</p><img src="/qimages/1215" /><p>d) For what length of race will each runner win? For what length of race will they tie?</p><p>How do your answers to part d) change if Tyrion&#39;s head start is</p> <ul> <li>doubled?</li> <li>cut in half?</li> </ul> <p>Suppose Mike went skiing six times over the winter season.</p> <ul> <li>Standard Rate:$50 per day and No registration fee.</li> <li>Frequent Extremist:: $40 per day and$100 registration fee.</li> </ul> <p>Suppose Mike went skiing 20 times over the winter season.</p><p>(a) How much would it cost him</p> <ul> <li>under the Standard Rate option?</li> <li>under the Frequent Extremist option?</li> </ul> <p>(b) Which option should Mike choose in this case? Explain.</p>
<p>Two fractions have denominators 3 and 4. Their sum is <code class='latex inline'>\frac{17}{12}</code>. If the numerators are switched, the sum is <code class='latex inline'>\frac{3}{2}</code>. Determine the two fractions. </p>
<p> The total of three sisters’ ages is 39. Dina is half as old as Michelle and 3 years younger than Juliette. How old are the sisters?</p>
<p>Suppose Mike went skiing six times over the winter season.</p> <ul> <li>Standard Rate: $50 per day and No registration fee.</li> <li>Frequent Extremist::$40 per day and $100 registration fee.</li> </ul> <p>Suppose Mike went skiing 20 times over the winter season.</p><p>Is there a scenario in which it does not matter which option Mike chooses? If so, describe it, referring to the graph.</p> <p>The demand function for a new product is <code class='latex inline'>p(x)=-4x + 42.5</code>, where <code class='latex inline'>x</code> is the quantity sold in thousands and p is the price in dollars. The company that manufactures the product is planning to buy a new machine for the plant. There are three different types of machine. The cost function for each machine is shown.</p> <ul> <li>Machine A: <code class='latex inline'>C(x) = 4.1x + 92.16</code></li> <li>Machine B: <code class='latex inline'>C(x) = 17.9x + 19.36</code></li> <li>Machine C: <code class='latex inline'>C(x) = 8.8x + 55.4</code></li> </ul> <p>Investigate the break-even quantities for each machine. Which machine would you recommend to the company?</p> <p>The sport of fencing has three main forms: sabre, foil, and épée. Sabre bouts take place within a rectangle of perimeter <code class='latex inline'> 52 \mathrm{~m} </code> . Decide whether the perimeter and each of the following pieces of information are sufficient to allow you to find the dimensions of the rectangle. Explain and justify your reasoning. Find the dimensions of the rectangle, where possible.</p><p>c) The length is twelve times the width.</p> <p>From his home in Point Alexander, Dan drove to Belleville at an average speed of <code class='latex inline'> 75 \mathrm{~km} / \mathrm{h} </code> . From her home in Chalk River, Ashley drove through Point Alexander to Belleville at an average speed of <code class='latex inline'> 85 \mathrm{~km} / \mathrm{h} </code> . The distance from Chalk River to Point Alexander is <code class='latex inline'> 18 \mathrm{~km} </code> . If Dan and Ashley left home at the same time,</p><p>a) after what length of time did Ashley overtake Dan?</p><p>b) how far were they from Point Alexander when Ashley overtook Dan?</p> <p>Playing tennis burns energy at a rate of about <code class='latex inline'> 25 \mathrm{~kJ} / \mathrm{min} </code> . Cycling burns energy at about <code class='latex inline'> 35 \mathrm{~kJ} / \mathrm{min} </code> . Hans exercised by playing tennis and then cycling. He exercised for 50 min altogether and used a total of <code class='latex inline'> 1450 \mathrm{~kJ} </code> of energy. For how long did he play tennis?</p> <p>A recording artist is offered two deals for her fourth CD release:</p> <ul> <li>Royalty only: <code class='latex inline'>\$1</code> per CD sold</li> <li><p>Partial royalty: <code class='latex inline'>\$2000</code> plus <code class='latex inline'>\$0.50</code> per CD sold</p></li> <li><p>Graph both linear relations on the same grid.</p></li> </ul>
<p>The sport of fencing has three main forms: sabre, foil, and épée. Sabre bouts take place within a rectangle of perimeter <code class='latex inline'> 52 \mathrm{~m} </code> . Decide whether the perimeter and each of the following pieces of information are sufficient to allow you to find the dimensions of the rectangle. Explain and justify your reasoning. Find the dimensions of the rectangle, where possible.</p><p>a) The width is <code class='latex inline'> 22 \mathrm{~m} </code> less than the length.</p>
<p>A boat took 5 h to travel 60 km up a river, against the current. The return trip took 3 h. </p><p>Find the speed of the boat in still water and the speed of the current.</p>
<p>The perimeter of a pool table is about <code class='latex inline'> 7.8 \mathrm{~m} </code> . Four times the length equals nine times the width. What are the dimensions of the table, in metres?</p>
<p>PROBLEM SOLVING A math test is worth 100 points and has 38 problems. Each problem is worth either 5 points or 2 points. How many problems of each point value are on the test?</p>
<p>Kwok is a hotel manager. His responsibilities include renting rooms for conferences. The hotel charges $250 per day plus$15 per person for the grand ballroom.</p><p>When trying to find out how many people can go when a total budget is given, is it better to use <code class='latex inline'>B= 14n + 250</code> or a rearranged version of this equation?</p> <ul> <li>Is the relationship between cost and number of guest linear or non-linear? Explain how you can tell.</li> </ul>
<p>Patrick owns an apartment building. He knows that the money he earns in a month depends on the rent he charges. This relationship can be modelled by <code class='latex inline'>E=\displaystyle{\frac{1}{50}}R(1650-R)</code>, where <code class='latex inline'>E</code> is Patrick&#39;s monthly earnings, in dollars, and <code class='latex inline'>R</code> is the amount of rent, in dollars, he charges each tenant.</p> <ul> <li>If Patrick wants to earn at least <code class='latex inline'>\$13 000</code>, between what two values should he set the rent?</li> </ul> <p>Natalie, Chantal, and Samara play together as a forward line on a hockey team. At the end of the season, Chantal had scored eight more goals than Natalie, while Samara had scored twice as many goals as Natalie. The three girls scored a total of 52 goals. How many goals did each girl score?</p> <ol> <li>Car wash The Outdoors Club held a car wash to raise money. They washed cars for <code class='latex inline'>\displaystyle \$ 5 </code> each and vans for <code class='latex inline'>\displaystyle \$7 </code> each. They washed 45 vehicles and earned <code class='latex inline'>\displaystyle \$ 243 . </code> How many of each type of vehicle did they wash?</li> </ol>
<p>Suppose Mike went skiing six times over the winter season.</p> <ul> <li>Standard Rate: $50 per day and No registration fee.</li> <li>Frequent Extremist:$40 per day and $100 registration fee.</li> </ul> <p>(a) How much would it cost him</p> <ul> <li>under the Standard Rate option?</li> <li>under the Frequent Extremist option?</li> </ul> <p>(b) Which option should Mike choose in this case? Explain.</p> <p>Some provinces have names with First Nations origins. For example, &quot;Ontario&quot; comes from an Iroquois word meaning “beautiful water&quot; or &quot;beautiful lake.&quot; If the number of provincial names with First Nations origins is <code class='latex inline'> a </code> , and the number with other origins is <code class='latex inline'> b </code> , the numbers are related by the following equations.</p><p><code class='latex inline'>\displaystyle a+b=10\\3 a-2 b=0 </code></p><p>a) Communication Interpret each equation in words.</p><p>b) Find the number of provinces that have names with First Nations origins.</p> <ol> <li>Car trip Maria drove from Owen Sound to Ottawa, a distance of <code class='latex inline'>\displaystyle 550 \mathrm{~km} . </code> The trip took 7 h. Maria drove at <code class='latex inline'>\displaystyle 70 \mathrm{~km} / \mathrm{h} </code> for part of the trip, and at <code class='latex inline'>\displaystyle 85 \mathrm{~km} / \mathrm{h} </code> for the remainder of the trip. How far did she drive at <code class='latex inline'>\displaystyle 70 \mathrm{~km} / \mathrm{h} </code> ?</li> </ol> <p>Ms. Frizzle has three daughters: Allison, Belle, and Claire. Today, January 1, their ages are, respectively,</p><p><code class='latex inline'> \displaystyle \begin{array}{cccccc} & A(n) = -(n + 30)+(2n + 5) \\ & B(n) = (7 -n)-(32 -2n) \\ & C(n) = (n - 26)-(n + 4) + (n - 3) \\ \end{array} </code></p><p>All ages are expressed in years, and n represents Ms. Flangan&#39;s age.</p><p>(c) How old was Ms. Flanagan when Cassandra was born?</p> <p>The Yankee Clipper leaves the pier at 9:00 A.M. at 8 knots (nautical miles per hour). A half hour later, The River Rover leaves the same pier in the same direction traveling at 10 knots. At what time will The River Rover overtake The Yankee Clipper?</p> <ol> <li>Canadian place names The two most common place names in Canada are Mount Pleasant and Centreville. The total number of places with these names is 31 . The number of places called Centreville is one less than the number of places called Mount Pleasant. This information can be modelled by the following linear system.</li> </ol> <p><code class='latex inline'>\displaystyle \begin{array}{l}m+c=31 \\ c=m-1\end{array} </code> Solve the system by substitution to find the number of places in Canada with each name.</p> <p>An online music download site offers two monthly plans:</p> <ul> <li>Plan A:$10 plus $1 per download</li> <li>Plan B:$1.50 per download</li> </ul> <p>a) Graph this linear system and find the solution.</p><p>b) Explain the conditions under which each plan is better.</p>
<p>The length of a picture frame is 3 in. greater than the width. The perimeter is less than 52 in. Describe the dimensions of the frame.</p>
<p>The Friendship Trail is a multi-use recreational trail that runs from Port Colborne to Fort Erie, a distance of 25 km. At 2 P.M., Debbie starts rollerblading from Port Colborne at 10 km/h. At the same time, Ken starts bicycling from Fort Erie at 20 km/h. When will they meet each other? How far from Fort Erie will they be at this time?</p>
<ol> <li>Buying bonds Li bought a Canada Savings Bond paying <code class='latex inline'>\displaystyle 5.5 \% </code> interest and a provincial government bond paying <code class='latex inline'>\displaystyle 6.5 \% </code> interest. She invested a total of <code class='latex inline'>\displaystyle \$15000 </code> and earned <code class='latex inline'>\displaystyle \$ 925 </code> in interest in the first year. How much did she pay for each bond?</li> </ol>
<p>Last summer, Betty earned $4200 by painting houses. She invested some of the money in a savings account that paid 3.5%/year and the rest in government bond that paid 4.5%/year. After one year, she has earned$174 in interest. How much did she invest at each rate?</p>
<p>There are 28 fish in an aquarium. There are eight more goldfish than neon tetras in the aquarium. How many goldfish are in the aquarium? How many neon tetras are in the aquarium?</p>
<p>Cersei and her brother Tyrion decide to race home. Cersei is a faster runner than Tyrion, so she gives him a head start. Their distance-time graphs are shown.</p><img src="/qimages/1215" /> <ul> <li>How much of a head start did Tyrion get?</li> </ul>
<p>Erica drove from Sarnia at <code class='latex inline'> 80 \mathrm{~km} / \mathrm{h} </code> . Aisha left Sarnia one hour later and drove along the same road at <code class='latex inline'> 100 \mathrm{~km} / \mathrm{h} </code> . How far from Sarnia did Aisha overtake Erica?</p>
<p>Sadia works at a theatre selling popcorn. Large boxes of popcorn sell for <code class='latex inline'>\displaystyle \$5.00 </code> and small boxes of popcorn sell for</p><p><code class='latex inline'>\displaystyle \$ 3.00 </code>. During her shift last night she sold 60 boxes of popcorn. Her receipts totalled <code class='latex inline'>\displaystyle \$260 </code>, not including taxes. a) How many large boxes of popcorn</p><p>did Sadia sell?</p><p>b) How many small boxes of popcorn</p><p>did she sell?</p> <p>When three numbers are added in pairs, the sums of the pairs are 22,39 , and 45 . What are the three numbers?</p> <p>There are 52 books in Lara Jean&#39;s library. There are 28 more fiction books than non-fiction books.</p><p>a) How many fiction books are in Lara Jean’s library?</p><p>b) How many non-fiction books are in Lara Jean’s library?</p> <p>A recording artist is offered two deals for her fourth CD release:</p> <ul> <li>Royalty only:$1 per CD sold</li> <li><p>Partial royalty: $2000 plus$0.50 per CD sold</p></li> <li><p>Sales figures for the artist&#39;s first three CDs are shown.</p></li> </ul> <img src="/qimages/1216" /><p>Which deal do you think the artist should choose? Explain your reasoning.</p>
<p> Write a word problem that can be solved using a system of linear equations and that has the solution <code class='latex inline'> (7,5) . </code> Have a classmate check that your problem gives the correct solution.</p>
<img src="/qimages/42842" /><p>Your car needs new brakes. You call a dealership and a local mechanic for prices.</p><p>Dealership &amp; <code class='latex inline'>\displaystyle \$24 </code> &amp; Labor cost per hour Local Mechanic &amp; <code class='latex inline'>\displaystyle \$ 45 </code> &amp; <code class='latex inline'>\displaystyle \$99 </code> </p><p>a. After how many hours are the total costs the same at both places? Justify your answer.</p><p>b. When do the repairs cost less at the dealership? at the local mechanic? Explain.</p> <p> The sum of the squares of two negative numbers is 74. The difference of their squares is 24. Determine the two numbers. </p> <p>Milk is leaking from a carton at a rate of 4 mL/min. There is 1500 ml. of milk in the carton at 8:30 a.m. </p><p>a) Write an equation and draw a graph for this situation.</p><p>b) Use your equation to determine algebraically when 1 L of milk will be left in the carton.</p> <p>To convert from centimetres to inches, you can use the fact that a 30 -cm ruler is just over 12 inches long. A person is <code class='latex inline'>\displaystyle 160 \mathrm{~cm} </code> tall. What is the person&#39;s approximate height, in inches?</p> <ol> <li>Resort costs A weekend at Bayview Lodge costs <code class='latex inline'>\displaystyle \$ 360 </code> and includes two nights&#39; accommodation and four meals. A week costs <code class='latex inline'>\displaystyle \$1200 </code> and includes seven nights&#39; accommodation and ten meals. If <code class='latex inline'>\displaystyle n </code> represents the cost of one night, and <code class='latex inline'>\displaystyle m </code> represents the cost of one meal, the relationship between the costs can be modelled by the following system of equations. <code class='latex inline'>\displaystyle \begin{array}{l}2 n+4 m=360 \\ 7 n+10 m=1200\end{array} </code> Determine the cost of one night and the cost of one meal.</li> </ol> <p>Use the diagram to find the values of <code class='latex inline'> x </code> and <code class='latex inline'> y </code> .</p><img src="/qimages/62313" /> <ol> <li>Tail wind A small plane took <code class='latex inline'>\displaystyle 3 \mathrm{~h} </code> to fly <code class='latex inline'>\displaystyle 960 \mathrm{~km} </code> from Ottawa to Halifax with a tail wind. On the return trip, flying into the wind, the plane took <code class='latex inline'>\displaystyle 4 \mathrm{~h} </code>. Find the wind speed and the speed of the plane in still air.</li> </ol> <ol> <li>Investments Zach invested in a term deposit that paid <code class='latex inline'>\displaystyle 4 \% </code> interest per annum and in a municipal bond that paid <code class='latex inline'>\displaystyle 6 \% </code> interest per annum. If he invested a total of <code class='latex inline'>\displaystyle \$ 13000 </code> and earned <code class='latex inline'>\displaystyle \$700 </code> interest in a year, how much did he invest at each rate?</li> </ol> <p>Sergio works for a a cellular phone company. He is paid$90/day plus $2.00 for each cellular phone that he sells. Aimee also works for the cell phone company, but she makes$120 per day and no extra money for selling cellular phones.</p><p>a) Write an equation to represent the amount that Savio earns in one day. Graph the equation.</p><p>b)) Write an equation to represent the amount that Aimee eams in one day. Graph this equation on the same grid you used in part a). How many cellular phones must Savio sell in order to make as much in a day as Aimee?</p>
<p>Profit A company manufactures and sells paddles. Its manufacturing costs are <code class='latex inline'> \$500 </code> , plus <code class='latex inline'> \$ 10 </code> per paddle. The company sells the paddles for <code class='latex inline'> \$18 </code> . The cost and revenue can be represented by the following system of equations.</p><p>Dollar Cost: <code class='latex inline'> \quad d=500+10 p </code> </p><p>Dollar Revenue: <code class='latex inline'>\displaystyle d=18 p </code></p><p>a) What does each variable represent?</p><p>b) Solve the system graphically.</p><p>c) How many paddles must be sold for the company to make a profit?</p> <p>Jobs You have a summer job at a car wash. You earn <code class='latex inline'>\displaystyle \$ 8.50 </code> per hour and are expected to pay a one-time fee of <code class='latex inline'>\displaystyle \$15 </code> for the uniform. If you work <code class='latex inline'>\displaystyle x </code> hours per week, how much will you make during the first week?</p> <p>Cersei and her brother Tyrion decide to race home. Cersei is a faster runner than Tyrion, so she gives him a head start. Their distance-time graphs are shown.</p><img src="/qimages/1215" /> <ul> <li>How fast does Tyrion run?</li> </ul> <p>Min likes to go cycling. He cycles partly on payed surfaces and partly off-road. through hilly and wooded areas. He cycles at 25 km/h on paved surfaces and at 10 km/h off-road. One day he cycled 41 km in 2 h. How far did he cycle off-road?</p> <p>These equations represent the total cost to charter buses from three different bus companies. In each equation, <code class='latex inline'>\displaystyle C </code> represents the total cost in dollars, and <code class='latex inline'>\displaystyle n </code> represents the number of passengers.</p><p>Company X: <code class='latex inline'>\displaystyle C=35 n+500 </code> </p><p>Company Y: <code class='latex inline'>\displaystyle C=25 n+2000 </code> </p><p>Company Z: <code class='latex inline'>\displaystyle C=37.50 n </code></p><p>Layton spends <code class='latex inline'>\displaystyle \$ 10150 </code> to charter buses to transport</p><p>326 passengers to a reunion.</p><p>a) Which bus company did Layton use?</p><p>b) Did Layton choose the best company if he wanted to spend as little as possible to charter the buses? Explain your answer.</p>
<p>Both equations in a linear system are written in the form Ax + By = C. Explain how you could predict the number of solutions using the coefficients and constants of the two equations.</p>
<p>The total number of states in Austria and Germany is 25 . Germany has 7 more states than Austria. Solve the following system of equations graphically to find the number of states in each country.</p><p><code class='latex inline'>\displaystyle a+g=25\\g=a+7 </code></p>
<p>PROBLEM SOLVING An investor owns shares of Stock A and Stock B. The investor owns a total of 200 shares with a total value of <code class='latex inline'>\displaystyle \$4000 </code>. How many shares of each stock does the investor own?</p><p><code class='latex inline'>\displaystyle \begin{array}{|c|c|}\hline Stock & Price \\ \hline A & \$ 9.50 \\ B & \\$ 27.00 \\ \hline\end{array} </code></p>
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