Question 638·Medium·Algebra

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In the $xy$ -plane, line $w$ passes through the points $(-4,3)$ and $(2,-9)$ . Which equation defines line $w$ ?

Two points with different $x$ -values determine one line. To write its equation in slope-intercept form, $y=mx+b$ , enter the points in a Desmos table and fit $y_1\sim mx_1+b$ . Desmos gives the slope $m$ and the $y$ -intercept $b$ . A given point’s $y$ -value isn’t necessarily the intercept; its $x$ -value must be $0$ .

Hints

Hint 1
An ordered pair $(x,y)$ keeps an $x$ -value with its matching $y$ -value. Put each given point in its own Desmos table row. Which $y$ -value goes with $x=2$ ?
Hint 2
In slope-intercept form $y=mx+b$ , $m$ is the slope and $b$ is the value of $y$ when $x=0$ . Fit both table rows, then use the two values under PARAMETERS to write the equation.

Step-by-step

Approach 1: Fit the line in Desmos
Step 1
Plot the two given points
Enter $(-4,3)$ and $(2,-9)$ as two rows in a Desmos table, with the $x$ -values under $x_1$ and the matching $y$ -values under $y_1$ . Desmos plots the points. As $x$ moves right from $-4$ to $2$ , $y$ moves down from $3$ to $-9$ , so the slope, the change in $y$ for each increase of $1$ in $x$ , is negative.
Step 2
Find the slope and intercept
Add $y_1\sim mx_1+b$ . The $\sim$ tells Desmos to fit a line to both rows. Under PARAMETERS, it shows $m=-2$ for the slope and $b=-5$ for the $y$ -intercept, the value of $y$ when $x=0$ . Only a point with $x=0$ gives you $b$ directly. Neither given point has that $x$ -value.
Step 3
Write the equation
Put Desmos’s values into slope-intercept form $y=mx+b$ . So line $w$ is defined by $y=-2x-5$ . Choice D.

Approach 2: Calculate the two parts by hand
Step 1
Find the slope from the changes
The slope is the change in $y$ divided by the change in $x$ . Subtract the coordinates in the same order on top and bottom: $\begin{aligned}m&=\frac{-9-3}{2-(-4)}\\&=\frac{-12}{6}\\&=-2.\end{aligned}$ The negative sign fits the points: $y$ falls as $x$ rises.
Step 2
Use a point to find the intercept
In $y=mx+b$ , $b$ is the $y$ -intercept. Use $(-4,3)$ and the slope $-2$ to find it. Substitute the point: $3=-2(-4)+b$ . Multiply: $3=8+b$ . Subtract $8$ from both sides: $-5=b$ .
Step 3
Combine the slope and intercept
The slope is $-2$ and the $y$ -intercept is $-5$ , so line $w$ is defined by $y=-2x-5$ . Choice D.

Question 638·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

Hints

Step-by-step

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