Question 509·Hard·Algebra

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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 $n$ ?

A linear system is solved where its two lines meet, because that point lies on both. If the crossing is unmarked but each line has two labeled points, use those points to write an equation for each line. Graph both equations in Desmos and click their intersection. Match the displayed decimals to exact fractions instead of rounding to nearby grid marks.

Hints

Hint 1
The slope of a line is its change in $y$ divided by its change in $x$ . Use points $A$ and $B$ in the same order in both parts of that fraction.
Hint 2
A point and a slope determine a line. Use point-slope form, $y-y_1=s(x-x_1)$ , with one labeled point on each line. Watch the signs when a coordinate is negative.
Hint 3
The solution is the intersection, not either of the labeled points. Once Desmos shows its coordinates, keep them in $(x,y)$ order and compare the decimals with the exact fractions in the choices.

Step-by-step

Build the lines and click their intersection
Step 1
Find the slope of line $m$
The slope is the change in $y$ divided by the change in $x$ . From $A(-4,5)$ to $B(5,-1)$ , type $\frac{-1-5}{5-(-4)}$ into Desmos. It shows $-0.666\ldots$ ; its fraction button gives $-\frac23$ . The negative slope fits a line that falls from left to right.
Step 2
Write the equation of line $m$
Use $A(-4,5)$ and the slope $-\frac23$ . Point-slope form says $y-y_1=s(x-x_1)$ for a line through $(x_1,y_1)$ , so line $m$ is:
$y-5=-\frac23(x+4).$
The $x+4$ comes from $x-(-4)$ .
Step 3
Find the slope of line $n$
From $C(-2,-1)$ to $D(4,5)$ , type $\frac{5-(-1)}{4-(-2)}$ into Desmos. It shows $1$ : $y$ rises by the same amount as $x$ .
Step 4
Write the equation of line $n$
Use $C(-2,-1)$ and slope $1$ in point-slope form. Line $n$ is:
$y+1=x+2.$
Both plus signs come from subtracting the negative coordinates of $C$ .
Step 5
Read the shared point
Type both equations into Desmos and click where the lines cross. Desmos labels the point approximately $(0.8,1.8)$ . Since $0.8=\frac45$ and $1.8=\frac95$ , the exact choice for the solution $(x,y)$ is $\left(\frac45,\frac95\right)$ . An intersection solves both lines, and its $x$ -coordinate comes first. Choice A.

Question 509·Hard·Algebra

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