The graph shows two lines, and , on a coordinate plane. Points , , , and are labeled on the lines. Which choice gives the solution to the system represented by lines and ?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 102 | Free SAT Prep | Aniko

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Question 102·Hard·Systems of Two Linear Equations in Two Variables

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The graph shows two lines, $m$ and $n$ , on a coordinate plane. Points $A$ , $B$ , $C$ , and $D$ are labeled on the lines.

Which choice gives the solution $(x,y)$ to the system represented by lines $m$ and $n$ ?

When a system is shown as two lines, use two clear points on each line to build the equations (slope from two points, then point-slope form). Convert at least one equation to $y=mx+b$ so substitution is quick, and be careful with negative slopes and distributing when you clear fractions; the intersection may look like a nearby grid point but can still be fractional.

Hints

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Use the labeled points

Each line has two labeled points. Use those points to find the slope of each line.

Write each line as an equation

After finding each slope, use point-slope form $y-y_1=m(x-x_1)$ to write an equation for each line.

Solve using substitution

Rewrite one equation as $y=\text{(something)}$ and substitute into the other equation to solve for $x$ , then find $y$ .

Desmos Guide

Enter the two-point equations for each line

In Desmos, enter an equation for line $m$ using points $A(-4,5)$ and $B(5,-1)$ :

y-5 = (-6/9)(x+4)

Then enter an equation for line $n$ using points $C(-2,-1)$ and $D(4,5)$ :

y+1 = (6/6)(x+2)

Find the intersection point

Click the intersection of the two lines. Desmos will show the intersection coordinates as decimals.

Match to an answer choice

Convert the decimal coordinates to fractions (for example, $0.8=\frac45$ and $1.8=\frac95$ ), then choose the option that matches those coordinates.

Step-by-step Explanation

Write an equation for line $m$ from two points

From the graph, line $m$ passes through $A(-4,5)$ and $B(5,-1)$ .

Its slope is

\frac{-1-5}{5-(-4)}=\frac{-6}{9}=-\frac23.

Using point-slope form with point $A$ :

\begin{aligned} y-5&=-\frac23(x+4) \\ 3y-15&=-2x-8 \\ 2x+3y&=7 \end{aligned}

Write an equation for line $n$ from two points

From the graph, line $n$ passes through $C(-2,-1)$ and $D(4,5)$ .

Its slope is

\frac{5-(-1)}{4-(-2)}=\frac66=1.

Using point-slope form with point $C$ :

\begin{aligned} y+1&=1(x+2) \\ y&=x+1 \end{aligned}

Substitute to create a one-variable equation

Substitute $y=x+1$ into $2x+3y=7$ :

\begin{aligned} 2x+3(x+1)&=7 \\ 5x+3&=7 \\ 5x&=4 \end{aligned}

Finish solving and select the ordered pair

Solve $5x=4$ to get $x=\frac45$ . Then use $y=x+1$ :

y=\frac45+1=\frac95.

So the solution to the system is $\left(\frac45,\frac95\right)$ .

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