Lectures and Quick Notes(1.5)
Chapter 1
Lectures and Quick Notes(1.5
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Lectures/Notes 6 Videos

Linear Word Problems Intro

We are now ready to try more various types of word problems that convert to linear systems!


  • Algebra
  • Substitution Method and/or
  • Elimination
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Section Intro

Intersection of Lines

As our first application, we are going to find the intersection point between two lines.

Solving the system of equations is equivalent to finding an intersection point between two lines!

ex Find the intersection point between the two lines below.

  • \displaystyle y = \frac{1}{2}x -1
  • \displaystyle y = -\frac{3}{4}x + 6

The two lines share the point of intersection say (a, b).

Since (a, b) is on both lines they can be now written in terms of a and b in the following way.

  • \displaystyle b = \frac{1}{2}a -1
  • \displaystyle b = -\frac{3}{4}a + 6

See video for detailed steps


\displaystyle \left(\frac{28}{5}, \frac{9}{5}\right)

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Intersection of Lines

Finding Intersection of Lines example

example: Find the point of intersection between

  • 4x+ y =3 and
  • -3x + 2y = -6

See Solution

\displaystyle (x, y) = (\frac{12}{11}, - \frac{15}{11})

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Intersection of Lines example

Distance Speed, Time Problem.

Let's think about how this will turn into a system of linear equations.

ex Dan is 0.5 km ahead of Arvin. Dan runs at 5km/h, Arvin runs at 7km/h.

a) Find Dan and Arvin's distance at a time, t.

D_d = 5t + 0.5, D_a= 7t

b) How long does it take for Avin to catch up to Dan?

It takes Arvin 15 mins to catch up to Dan.

c) How long does it take for Arvin to be 2 km ahead of Dan?

It takes Arvin 1 hr & 15 mins to be 2 km ahead of Dan.

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Solve a Distance Speed, Time Problem

Running Opposite direction on Track

ex Abe and Ben runs around a track. They start is the same but Abe runs clock-wise while Ben runs counter clock-wise. Abe can run at 3km/hr and Ben runs at 5km/hr. If the track is 10 km, how long does it take for them to meet?

Therefore, they meet in 1 hour and 15 mins.

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Running Opposite direction on Track

Mixture Problem with One Unknown ver

Here is a type of word problem that pretty much confuses every student when they see it for the first time: Mixture Problem!

ex John mixes 2L of 3-% concentrate orange juice with another that is 50%. In the resulting orange juice mixture the percentage of orange juice mixture the percentage of orange is 34%. How much of the 50% orange juice was mixed?

Therefore, 7.75 L 50% juice needs to be taken.

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Mixture Problem ex1