Question 234·Medium·Algebra

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In the $xy$ -plane, the two bold lines shown represent a system of linear equations.

Which choice gives the solution $(x,y)$ to the system?

When a system is shown as two lines, the solution is their intersection. If the intersection is not clearly at a grid point, use two labeled points on each line to write each equation (find slope, then $y=mx+b$ ), and solve the resulting system algebraically.

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Remember what the solution represents

The solution to a system is the point where the two lines cross.

Use the labeled points

Pick the two labeled points on each line to find that line’s slope and equation.

Set the equations equal

Once you have both equations in the form $y=\dots$ , set them equal and solve for $x$ , then substitute back to get $y$ .

Desmos Guide

Enter the two lines

From the labeled points, determine the equations $y=x+1$ and $y=-\frac{1}{2}x+3$ , then type both equations into Desmos.

Find the intersection

Click the point where the two lines intersect. Read the intersection’s coordinates from the label Desmos shows.

Step-by-step Explanation

Write an equation for each line from the graph

From the graph, one line passes through $(-2,-1)$ and $(4,5)$ . Its slope is

\frac{5-(-1)}{4-(-2)}=\frac{6}{6}=1

So its equation is $y=x+1$ (since $-1=-2+b \Rightarrow b=1$ ).

The other line passes through $(-2,4)$ and $(4,1)$ . Its slope is

\frac{1-4}{4-(-2)}=\frac{-3}{6}=-\frac{1}{2}

So its equation is $y=-\frac{1}{2}x+3$ (since $4=1+ b \Rightarrow b=3$ ).

Solve the system by setting the expressions for $y$ equal

Set $x+1$ equal to $-\frac{1}{2}x+3$ :

\begin{aligned} x+1 &= -\frac{1}{2}x+3 \\ \frac{3}{2}x &= 2 \\ x &= \frac{4}{3} \end{aligned}

Find $y$ using either equation

Substitute $x=\frac{4}{3}$ into $y=x+1$ :

y=\frac{4}{3}+1=\frac{7}{3}

So the solution (intersection point) is $\left(\frac{4}{3},\frac{7}{3}\right)$ .

Question 234·Medium·Algebra

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation