In the -plane, line passes through the point and is perpendicular to the line . Which equation represents line ?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 93 | Free SAT Prep | Aniko

00:00

Question 93·Hard·Linear Equations in Two Variables

0 / 140

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-line questions, immediately find the slope of the given line by solving for $y$ and reading off $m$ in $y = mx + b$ , then take the negative reciprocal for the perpendicular slope. Use point-slope form with the given point to write the new line, and, if answer choices are in standard form $Ax + By = C$ , quickly rearrange your equation to match that format. As a shortcut with multiple-choice options, you can also check each choice’s slope (by isolating $y$ ) and test whether it passes through the given point; the correct answer must have the perpendicular slope and satisfy the point coordinates.

Hints

Click me!

Identify the slope of the given line

First, solve the equation $3x + 4y = 1$ for $y$ to write it in the form $y = mx + b$ . What is the slope $m$ of this line?

Relate perpendicular slopes

Once you know the slope of the given line, remember: the slope of a line perpendicular to it is the negative reciprocal. How do you find the negative reciprocal of a fraction?

Use the given point

Use point-slope form, $y - y_1 = m(x - x_1)$ , with the perpendicular slope and the point $(5,-2)$ to write the equation of line $\ell$ before comparing to the answer choices.

Match the form of the answer choices

The answer choices are all in the form $Ax + By = C$ . After you find an equation for the line, rearrange it into that form and then see which option it matches.

Desmos Guide

Graph the original line and the point

Type 3x + 4y = 1 into Desmos to graph the given line, and type (5, -2) to plot the point that line $\ell$ must pass through.

Graph each answer choice

Enter each option as a separate equation in Desmos: 3x + 4y = 26, 4x - 3y = 26, 3x - 4y = 26, and 4x + 3y = -26. All four lines will appear on the graph.

Visually check perpendicularity and the point

Look for the line that both passes exactly through the point $(5,-2)$ and forms a right angle (perpendicular) with the graph of 3x + 4y = 1. The equation of that line is the correct choice.

Step-by-step Explanation

Find the slope of the given line

Rewrite the equation of the given line in slope-intercept form ( $y = mx + b$ ) to see its slope.

Start with:

3x + 4y = 1

Solve for $y$ :

4y = -3x + 1

y = -\frac{3}{4}x + \frac{1}{4}

So the slope of the given line is $-3/4$ .

Use the perpendicular slope relationship

Slopes of perpendicular lines are negative reciprocals of each other.

The negative reciprocal of $-3/4$ is $4/3$ (flip the fraction and change the sign).

So the slope of line $\ell$ must be $4/3$ .

Write an equation using the point

Use point-slope form with slope $4/3$ and point $(5,-2)$ :

y - (-2) = \frac{4}{3}(x - 5)

This simplifies to:

y + 2 = \frac{4}{3}x - \frac{20}{3}

Subtract $2$ (which is $6/3$ ) from both sides:

y = \frac{4}{3}x - \frac{20}{3} - 2 = \frac{4}{3}x - \frac{26}{3}

So one form of the equation of line $\ell$ is $y = \frac{4}{3}x - \frac{26}{3}$ .

Convert to standard form and match a choice

Convert $y = \frac{4}{3}x - \frac{26}{3}$ to the standard form $Ax + By = C$ .

Multiply both sides by $3$ to clear denominators:

3y = 4x - 26

Subtract $3y$ from both sides (or move terms) to get:

4x - 3y = 26

This matches choice B, so the equation of line $\ell$ is $4x - 3y = 26$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation