Now You Try

<p>Solve the following systems of equations, and explain the nature of each intersection:</p><p><code class='latex inline'>\displaystyle
\begin{array}{lllll}
&2x + y &= 3 \\
&2x + y &= 4 \\
\end{array}
</code></p>

<p> Consider this system of linear equations:<br>
<code class='latex inline'>\displaystyle
\begin{array}{cccc}
& 2x + y = 7\\
&8x + 4y = 10\\
\end{array}
</code> </p><p><strong>a)</strong> Can you create an equivalent system that contains only one variable? </p><p><strong>b)</strong> What does your result for part a) suggest about the solution to the original system? </p><p><strong>c)</strong> What does your result for part a) suggest about the graphs of both lines? </p>

<p>Explain how you know that this system of equations has no solution.</p><p><code class='latex inline'>\displaystyle
\begin{array}{lllll}
&15 -6y = 9x\\
&3x + 2y = 8
\end{array}
</code></p>

<p>For the first equation in the system of linear equations below, write an equivalent equation without denominators. Then solve the system.</p><p><code class='latex inline'>\displaystyle \left\{\begin{array}{l}\frac{x}{5}+\frac{y}{3}=6 \\ x-2 y=8\end{array}\right. </code></p>

<p>Use the equation <code class='latex inline'>3x + 4y= 2</code></p><p>Write another equation that will create a linear system with each number of solutions.</p><p>i) none</p><p>ii) one</p><p>iii) infinitely many</p>

<p>Without graphing, determine whether each system has one solution,
no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle
x+4 y=8\\y+2 x=0
</code></p>

<p>Solve the following systems of equations, and explain the nature of each intersection:</p><p><code class='latex inline'>\displaystyle
\begin{array}{lllll}
&7x -3y &= 9 \\
&35x - 15y &= 45
\end{array}
</code></p>

<p>a) Can you solve the linear system <code class='latex inline'>y=3x-2</code> and <code class='latex inline'>6x-2y-4=0</code>?
Explain your reasoning.</p><p>b) Can you solve the linear system <code class='latex inline'>y=4x-3</code> and <code class='latex inline'>8x-2y+ 5 =0</code>?</p><p>Explain your reasoning.</p><p>c) Explain how you can tell, without
solving, how many solutions a linear system has.</p>

<p>Without graphing, determine whether each system has one solution,
no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle
2 y-x-4=0\\3 x-6 y-12=0
</code></p>

<p>Without graphing, determine whether each system has one solution,
no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle
\begin{aligned} & 3 c+d=4 \\ & 6 c+2 d=8 \end{aligned}
</code></p>

<p>Solve each system of equations by elimination. Check each solution.</p><p><code class='latex inline'>\displaystyle 2 x+5 y=3 </code></p><p><code class='latex inline'>\displaystyle 4 x+10 y-6=0 </code></p>

<p>Without graphing, determine whether each system has one solution,
no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle
x+y=2\\3 x=6-3 y
</code></p>

<p> Without graphing, determine whether each system has one solution,
no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle
3 x-y=0\\6 x-2 y=3
</code></p>

<p> Write an equation that forms a system of equations with <code class='latex inline'> x+y=4 </code> , so
that the system has</p><p>a) no solution</p><p>b) infinitely many solutions</p><p>c) one solution</p>

<p>Without graphing, determine whether each system has one solution,
no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle
x+2 y-7=0\\3 x+6 y-14=0
</code></p>

<p> Write a system of equations that has the point <code class='latex inline'> (3,2) </code> as</p><p>a) the only solution</p><p>b) one of infinitely many solutions</p>

<p>Without graphing, determine whether each system has one solution,
no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle
2 y=3 x-1\\8 y-4=12 x
</code></p>

<p>Solve each system of equations by elimination. Check each solution.</p><p><code class='latex inline'>\displaystyle 2 x-3 y=8 </code></p><p><code class='latex inline'>\displaystyle 4 x-6 y=10 </code></p>

<p>Without graphing, determine whether each system has one solution
no solution, or infinitely many solutions.</p><p><code class='latex inline'>\displaystyle
4 x-2 y=0\\2 x-y=3
</code></p>