Question 443·Medium·Algebra

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A line in the $xy$ -plane passes through the point $(-3,\,7)$ and has a slope of $4$ . The equation of the line can be written in the form $y = mx + b$ , where $m$ and $b$ are constants.

What is the value of $b$ ?

Your answer

A slope and a point determine a line in slope-intercept form, $y=mx+b$ . The slope gives $m$ ; put the point’s coordinates into the equation to find $b$ . A Desmos regression can solve the resulting equation. The point’s $y$ -coordinate is not automatically $b$ : it equals $b$ only when the point’s $x$ -coordinate is $0$ .

Hints

Hint 1
In slope-intercept form, $b$ is the value of $y$ when $x=0$ . Check the point’s $x$ -coordinate before treating its $y$ -coordinate as the intercept.
Hint 2
A point on a line must make its equation true. Replace both $x$ and $y$ with the point’s coordinates, and replace $m$ with the given slope. What equation does that leave for $b$ ?

Step-by-step

Put the point into slope-intercept form
Step 1
Identify what $b$ measures
The $y$ -intercept is where the line meets the $y$ -axis, so $x=0$ . In $y=mx+b$ , setting $x=0$ gives $y=m(0)+b=b$ . The given point has $x=-3$ , not $0$ , so its $y$ -coordinate $7$ isn’t automatically $b$ .
Step 2
Turn the slope and point into an equation
The point $(-3,7)$ means the line gives $y=7$ when $x=-3$ , and its slope gives $m=4$ . Put both coordinates into $y=mx+b$ : $7=4(-3)+b$ . Keep the negative sign on $-3$ .
Step 3
Solve for the intercept
Type $7\sim4(-3)+b$ as a regression. The $\sim$ tells Desmos to find the value that makes both sides match. Under PARAMETERS, Desmos shows $b=19$ . So the requested value is $19$ . Grid in 19.

Question 443·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

Lessons that teach this