Question 428·Medium·Algebra

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In the $xy$ -plane, line $\ell$ passes through the points $(-3,2)$ and $(5,-2)$ . Which equation represents line $\ell$ ?

Two points determine a line's slope-intercept form, $y=mx+b$ . Put the points in a Desmos table, then use a linear regression to find the slope $m$ and the $y$ -intercept $b$ . A point's $y$ -value is the intercept only when its $x$ -value is $0$ . Check that the slope's sign matches whether the points rise or fall from left to right.

Hints

Hint 1
Keep each point paired: enter its $x$ -coordinate and $y$ -coordinate in the same row of a Desmos table. As you read from the first point to the second, does $y$ rise or fall as $x$ increases?
Hint 2
In slope-intercept form, $y=mx+b$ , $m$ tells you how $y$ changes as $x$ increases, while $b$ is the value of $y$ when $x=0$ . Fit the two table rows to that form.

Step-by-step

Approach 1: Fit the two points in Desmos
Step 1
Pair the coordinates in a table
Enter the points as paired rows in a Desmos table: put $-3$ beside $2$ , and $5$ beside $-2$ . Reading down the table, $x$ increases while $y$ decreases, so the line's slope must be negative: it falls as you move right.
Step 2
Read the slope and intercept
Type $y_1\sim mx_1+b$ below the table. The $\sim$ tells Desmos to fit $y=mx+b$ to the points, where $m$ is the slope and $b$ is the $y$ -intercept, the value when $x=0$ . Under PARAMETERS, Desmos reports $m=-0.5$ and $b=0.5$ . So line $\ell$ is $y=-\frac12x+\frac12$ . The positive intercept is not the $y$ -coordinate of either given point. Choice B.

Approach 2: Build the equation from the slope
Step 1
Find the slope from the changes
A line's slope is the change in $y$ divided by the matching change in $x$ . Subtract the coordinates in the same order on top and bottom: $m=\frac{-2-2}{5-(-3)}$ . Type that fraction in Desmos; it gives $-0.5$ , which equals $-\frac12$ . The negative sign agrees with the falling line.
Step 2
Use a point to find what is missing
In $y=mx+b$ , the intercept $b$ is the value of $y$ when $x=0$ . The point $(-3,2)$ has $x=-3$ , so its $y$ -value isn't $b$ . Put that point and the slope into the equation to find $b$ : $2=-\frac12(-3)+b$ .
Step 3
Solve for the intercept
Type $2\sim-\frac12(-3)+b$ so Desmos solves the equation for $b$ . Under PARAMETERS, it reports $b=0.5$ . With slope $-\frac12$ and intercept $\frac12$ , line $\ell$ is $y=-\frac12x+\frac12$ . Choice B.

Question 428·Medium·Algebra

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