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Math  /  Algebra

QuestionFind the slope of a line parallel and perpendicular to the line.
10. 3y=−4x−243y = -4x - 24 || slope = ______ ⊥ slope = ______

Studdy Solution

STEP 1

What is this asking? We're finding the slopes of lines parallel and perpendicular to a given line. Watch out! Don't mix up parallel and perpendicular slopes!

STEP 2

1. Rewrite the equation
2. Find the slope
3. Find the parallel slope
4. Find the perpendicular slope

STEP 3

We want to rewrite the equation 3y=−4x−243y = -4x - 24 in slope-intercept form (y=mx+by = mx + b), where mm is the slope and bb is the y-intercept.
To do this, we need to isolate yy.
Let's **divide both sides** of the equation by 3\textbf{3}: 3y3=−4x−243 \frac{3y}{3} = \frac{-4x - 24}{3} y=−4x3−243 y = \frac{-4x}{3} - \frac{24}{3} y=−43x−8 y = -\frac{4}{3}x - 8

STEP 4

Now, our equation is in slope-intercept form!
Remember, the **slope** is the coefficient of xx.
In our equation y=−43x−8y = -\frac{4}{3}x - 8, the **slope** is −43-\frac{\textbf{4}}{\textbf{3}}.

STEP 5

Parallel lines have the *same* slope.
Since the slope of our original line is −43-\frac{4}{3}, the slope of a line parallel to it is *also* −43-\frac{\textbf{4}}{\textbf{3}}.

STEP 6

Perpendicular lines have negative reciprocal slopes.
To find the negative reciprocal of a fraction, we flip the fraction and change the sign.
The negative reciprocal of −43-\frac{4}{3} is 34\frac{\textbf{3}}{\textbf{4}}.
So, the slope of a line perpendicular to our original line is 34\frac{3}{4}.

STEP 7

Parallel slope: −43-\frac{4}{3} Perpendicular slope: 34\frac{3}{4}

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