Question 404·Hard·Algebra

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The graph shows two distinct lines, $p$ and $q$ , on the $xy$ -plane. Each line passes through two labeled points.

Which choice gives the exact coordinates of the intersection point of lines $p$ and $q$ ?

An intersection is a point that lies on both lines. With two labeled points per line, find each slope, then use a point and its slope to write each line’s equation. Graph the equations in Desmos and click their crossing. A graph label can be rounded, so check any proposed exact coordinates in both equations.

Hints

Hint 1
A line’s slope is the change in $y$ divided by the change in $x$ . Use A and B in the same order on top and bottom. What slope do their coordinates give for line $p$ ?
Hint 2
A point-slope equation uses one point and the slope: $y-y_1=m(x-x_1)$ . Using A, the horizontal part is $x-(-5)=x+5$ . Keep the negative slope because line $p$ falls as you move right.
Hint 3
An intersection makes both line equations true. Desmos gives a decimal label for the crossing, but the question wants exact fractions. How can you turn its displayed $x$ -coordinate into a fraction and check it in both equations?

Step-by-step

Graph equations built from the labeled points
Step 1
Find the slope of line p
The figure places $A=(-5,3)$ and $B=(1,-1)$ on $p$ . Slope is the change in $y$ divided by the change in $x$ . Type $\frac{-1-3}{1-(-5)}$ in Desmos, keeping the points in the same order. Desmos shows $-0.666\ldots$ ; its fraction button shows $-\frac{2}{3}$ .
Step 2
Graph line p
Use A and the slope in point-slope form, $y-y_1=m(x-x_1)$ . Type $y-3=-\frac{2}{3}(x+5)$ . Desmos draws the line through A and B. The $x+5$ is $x-(-5)$ , the horizontal change from A.
Step 3
Find the slope of line q
The figure places $C=(-1,5)$ and $D=(3,-1)$ on $q$ . Type their change in $y$ over change in $x$ , $\frac{-1-5}{3-(-1)}$ . Desmos shows $-1.5$ ; its fraction button shows $-\frac{3}{2}$ .
Step 4
Click the intersection
Use C and its slope to type $y-5=-\frac{3}{2}(x+1)$ . Desmos draws $q$ . Click where the two lines cross; the label shows $(4.6,-3.4)$ . That point is on both lines, but its decimal label alone does not establish exact fractions.
Step 5
Turn the displayed x-coordinate into a fraction
The label’s $x$ -coordinate is $4.6$ , which could be rounded, so treat it as a candidate. Since it has one decimal place, write it over $10$ :
$4.6=\frac{46}{10}$
Reduce by $2$ :
$\frac{46}{10}=\frac{23}{5}.$
Step 6
Check the exact coordinates in both lines
Test $x=\frac{23}{5}$ in both line rules. Type $3-\frac{2}{3}(\frac{23}{5}+5)$ and $5-\frac{3}{2}(\frac{23}{5}+1)$ on new Desmos lines. Each prints $-3.4$ ; each result’s fraction button shows $-\frac{17}{5}$ . So the same exact point satisfies both line equations. Their intersection is $\left(\frac{23}{5},-\frac{17}{5}\right)$ . Choice A.

Question 404·Hard·Algebra

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