Question 463·Hard·Algebra

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In the $xy$ -plane, line $\ell$ passes through the point $(5,-2)$ and is perpendicular to the line $3x + 4y = 1$ . Which equation represents line $\ell$ ?

For perpendicular lines, the slopes are negative reciprocals: flip the fraction and change its sign. Graph the given equation in Desmos to check which way it runs, then use its coefficients to find the exact slope. Build a line with the perpendicular slope and substitute the given point to find its constant. Changing only the sign is not enough.

Hints

Hint 1
In standard form $Ax+By=C$ , the slope is $-A/B$ , not $A$ or $C$ . Graph the given line in Desmos to check its direction, then use the coefficients of $x$ and $y$ to find the exact slope.
Hint 2
The negative reciprocal of $a/b$ is $-b/a$ : flip the fraction and change its sign. Perpendicular lines have slopes related this way. What slope does the new line need?
Hint 3
A point on a line makes its equation true. Once you have the new line’s slope, write an equation with an unknown constant. What happens when you substitute the given point’s $x$ - and $y$ -coordinates?

Step-by-step

Find the perpendicular slope, then use the point
Step 1
Check the given line’s direction
Type $3x+4y=1$ in Desmos. The graph falls as $x$ increases, so its slope, the change in $y$ for each $1$ added to $x$ , is negative.
Step 2
Find the exact slope
In standard form $Ax+By=C$ , the slope is $-A/B$ . The given equation has $A=3$ and $B=4$ , so its exact slope is $-\frac{3}{4}$ .
Step 3
Turn to the perpendicular slope
Line $\ell$ is perpendicular, so it meets the given line at a right angle. A direction of right $4$ , down $3$ turns into right $3$ , up $4$ . That gives slope $\frac{4}{3}$ : flip the slope fraction and change its sign. Changing only the sign would give $\frac{3}{4}$ , which is not perpendicular.
Step 4
Write the new line with an unknown constant
For $Ax+By=C$ , the slope is $-A/B$ . Choosing $A=4$ and $B=-3$ gives the needed slope $\frac{4}{3}$ , so write $4x-3y=C$ . Here $C$ is the unknown constant that places the line through the point.
Step 5
Substitute the point
Because $(5,-2)$ lies on line $\ell$ , its coordinates must make the new equation true. Put $5$ in for $x$ and $-2$ in for $y$ , in that order: $4(5)-3(-2)=C$ . Keep both minus signs in $-3(-2)$ .
Step 6
Find the constant and finish the equation
Type $4(5)-3(-2)$ in Desmos. It shows $26$ , so the constant is $26$ and line $\ell$ is $4x-3y=26$ . Choice B.

Question 463·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this