Question 413·Medium·Algebra

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In the $xy$ -plane, line $s$ passes through the point $(6,-2)$ and is perpendicular to the line given by $4x - 5y = 20$ . What is the $y$ -intercept of line $s$ ?

A line equation, a point on a perpendicular line, and a request for an intercept give you three jobs. Find the original slope by graphing its equation in Desmos and reading two points. Flip the slope’s fraction and change its sign for the new line, then use the given point to find its $y$ -intercept. The point’s $y$ -coordinate is not automatically the intercept.

Hints

Hint 1
A slope measures how far a line rises for each move to the right. Graph the given line and read its crossings with the axes. How much does $y$ rise between those crossings, and how far does $x$ move?
Hint 2
For perpendicular lines, the slopes are negative reciprocals: flip the fraction and change its sign. Changing only the sign, or only flipping the fraction, won’t give the new line the right slope.
Hint 3
In $y=mx+b$ , $b$ is the $y$ -intercept, where $x=0$ . The given point has a different $x$ -coordinate, so put both of its coordinates into the equation and solve for $b$ .

Step-by-step

Graph the given line, then find the new intercept
Step 1
Read two points on the given line
Type $4x-5y=20$ in Desmos. Click the line, then its axis crossings. Desmos shows $(0,-4)$ and $(5,0)$ . These are points on the given line, not on line $s$ .
Step 2
Find the given line’s slope
From $(0,-4)$ to $(5,0)$ , $y$ rises $4$ while $x$ moves right $5$ . A slope is rise divided by run, so the given line’s slope is $\frac45$ .
Step 3
Get the perpendicular slope
Line $s$ is perpendicular to the given line. Perpendicular slopes flip the fraction and change its sign. So $\frac45$ becomes $-\frac54$ , the slope of line $s$ .
Step 4
Use the point to make an equation
In slope-intercept form, $y=mx+b$ , the number $b$ is the $y$ -intercept. The point $(6,-2)$ means $x=6$ gives $y=-2$ , so substitute it and line $s$ ’s slope: $-2=-\frac54(6)+b$ . The point’s $-2$ is not the intercept because its $x$ -coordinate isn’t $0$ .
Step 5
Solve for the intercept
Type $-2\sim-\frac54(6)+b$ in Desmos. The $\sim$ tells Desmos to find $b$ ; it reports $b=5.5$ under PARAMETERS. Type $b$ on the next line and use the fraction button to see $\frac{11}{2}$ . So line $s$ has $y$ -intercept $\frac{11}{2}$ . Choice D.

Question 413·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this