Question 388·Hard·Algebra

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In the $xy$ -plane, line $k$ is defined by $4x - 5y = 20$ and line $j$ is defined by $y = -2x + 3$ . The two lines intersect at point $P$ . Line $m$ is perpendicular to line $k$ and passes through point $P$ . Which of the following is an equation of line $m$ ?

A line through the intersection of two other lines needs a point and a slope. Graph the given equations in Desmos and click their intersection for the point. For perpendicular lines, flip the reference line’s slope and change its sign, then use the point to fix the new line’s position. Check both the slope and the point.

Hints

Hint 1
An intersection lies on both lines, so its coordinates satisfy both equations. Graph the given lines and click where they cross. How will that point help you place line $m$ ?
Hint 2
In standard form $Ax+By=C$ , a line’s slope is $-\frac{A}{B}$ . Use the coefficients of line $k$ , then remember that a perpendicular slope needs both a flip and a sign change.
Hint 3
A slope fixes a line’s direction, but not its position. Write an equation with the perpendicular slope and an unknown constant. Then put both coordinates of $P$ into it to find that constant.

Step-by-step

Find the intersection, then build the perpendicular line
Step 1
Find the point line $m$ must pass through
Type $4x-5y=20$ and $y=-2x+3$ on separate Desmos lines. Click their intersection: Desmos shows $P=(2.5,-2)$ . An intersection is on both lines, so this is the point line $m$ must pass through.
Step 2
Read line $k$ ’s slope
For a line in standard form $Ax+By=C$ , the slope is $-\frac{A}{B}$ : it’s the coefficient of $x$ when you solve for $y$ . Line $k$ has $A=4$ and $B=-5$ , so its slope is $-\frac{4}{-5}=\frac45$ . The $20$ sets the line’s position, not its slope.
Step 3
Get the perpendicular slope
Perpendicular lines meet at a right angle, so their slopes are negative reciprocals: flip the fraction and change its sign. From line $k$ ’s slope $\frac45$ , line $m$ needs slope $-\frac54$ . Changing only the sign would leave the wrong slope.
Step 4
Write a line with that slope
In standard form, $5x+4y=c$ has slope $-\frac54$ , whatever the constant $c$ is. So write $5x+4y=c$ . This gives line $m$ the right direction; the point $P$ will determine $c$ .
Step 5
Use $P$ to place the line
Because $P=(2.5,-2)$ lies on line $m$ , put its coordinates into $5x+4y=c$ . Type $5(2.5)+4(-2)$ on the next Desmos line. It shows $4.5$ ; its fraction button gives $\frac92$ . So $c=\frac92$ , and line $m$ is $5x+4y=\frac92$ .
Step 6
Match the equation to the choices
Line $m$ is $5x+4y=\frac92$ . Multiply every term by $2$ to clear the fraction: $10x+8y=9$ . This equation has the perpendicular slope and passes through $P$ , so it defines line $m$ . Choice D.

Question 388·Hard·Algebra

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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