Math / Algebra

QuestionFind the equation of the line that contains the point $(-2,-1)$ and is perpendicular to the line $2 x+3 y=9$ . Write the line in slope-intercept form, if possible. Graph the lines.
Select the correct choice below and fill in the answer box to complete your choice. A. The equation of the perpendicular line in slope-intercept form is $\square$ $\square$ . (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.) B. The equation of the perpendicular line cannot be written in slope-intercept form. The equation of the perpendicular line is $\square$ $\square$ . (Simplify your answer. Use integers or fractions for any numbers in the equation.)

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that passes through a specific point and is perpendicular to another given line, then write the equation in slope-intercept form if we can. Watch out! Remember that perpendicular lines have negative reciprocal slopes!
Don't forget to simplify your final equation.

STEP 2

1. Find the slope of the given line.
2. Find the slope of the perpendicular line.
3. Find the equation of the perpendicular line.

STEP 3

We're given the equation $2x + 3y = 9$ .
To find the slope, let's rewrite this equation in slope-intercept form ( $y = mx + b$ ), where $m$ is the slope and $b$ is the y-intercept.
First, we want to isolate $y$ .
Subtract $2x$ from both sides: $3y = -2x + 9$ .
Now, divide both sides by $3$ to get $y$ by itself: $y = -\frac{2}{3}x + 3$ .

STEP 4

Now, we can easily see that the slope of the given line is $-\frac{2}{3}$ .
Awesome!

STEP 5

Remember, perpendicular lines have negative reciprocal slopes.
So, we need to flip the fraction and change the sign.
The negative reciprocal of $-\frac{2}{3}$ is $\frac{3}{2}$ .
This is the slope of our perpendicular line.

STEP 6

We know the slope of our perpendicular line is $\frac{3}{2}$ and it passes through the point $(-2, -1)$ .
We can use the point-slope form of a linear equation: $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the given point.
Plugging in our values, we get $y - (-1) = \frac{3}{2}(x - (-2))$ .

STEP 7

Let's simplify! $y + 1 = \frac{3}{2}(x + 2)$ .
Distribute the $\frac{3}{2}$ : $y + 1 = \frac{3}{2}x + \frac{3}{2} \cdot 2$ , which simplifies to $y + 1 = \frac{3}{2}x + 3$ .

STEP 8

Finally, subtract $1$ from both sides to get $y$ by itself: $y = \frac{3}{2}x + 3 - 1$ , which simplifies to $y = \frac{3}{2}x + 2$ .
This is our equation in slope-intercept form!

STEP 9

A. The equation of the perpendicular line in slope-intercept form is $y = \frac{3}{2}x + 2$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the slope of the given line.
2. Find the slope of the perpendicular line.
3. Find the equation of the perpendicular line.

STEP 3

STEP 4

Now, we can easily see that the slope of the given line is $-\frac{2}{3}$ .
Awesome!

STEP 5

Remember, perpendicular lines have negative reciprocal slopes.
So, we need to flip the fraction and change the sign.
The negative reciprocal of $-\frac{2}{3}$ is $\frac{3}{2}$ .
This is the slope of our perpendicular line.

STEP 6

STEP 7

Let's simplify! $y + 1 = \frac{3}{2}(x + 2)$ .
Distribute the $\frac{3}{2}$ : $y + 1 = \frac{3}{2}x + \frac{3}{2} \cdot 2$ , which simplifies to $y + 1 = \frac{3}{2}x + 3$ .

STEP 8

Finally, subtract $1$ from both sides to get $y$ by itself: $y = \frac{3}{2}x + 3 - 1$ , which simplifies to $y = \frac{3}{2}x + 2$ .
This is our equation in slope-intercept form!

STEP 9

A. The equation of the perpendicular line in slope-intercept form is $y = \frac{3}{2}x + 2$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the slope of the given line.2. Find the slope of the perpendicular line.3. Find the equation of the perpendicular line.

STEP 3

STEP 4

Now, we can easily see that the **slope** of the given line is −23-\frac{2}{3}−32​.Awesome!

STEP 5

Remember, perpendicular lines have **negative reciprocal slopes**.So, we need to flip the fraction and change the sign.The negative reciprocal of −23-\frac{2}{3}−32​ is 32\frac{3}{2}23​.This is the **slope** of our perpendicular line.

STEP 6

STEP 7

Let's simplify! y+1=32(x+2)y + 1 = \frac{3}{2}(x + 2)y+1=23​(x+2).Distribute the 32\frac{3}{2}23​: y+1=32x+32⋅2y + 1 = \frac{3}{2}x + \frac{3}{2} \cdot 2y+1=23​x+23​⋅2, which simplifies to y+1=32x+3y + 1 = \frac{3}{2}x + 3y+1=23​x+3.

STEP 8

Finally, subtract 111 from both sides to get yyy by itself: y=32x+3−1y = \frac{3}{2}x + 3 - 1y=23​x+3−1, which simplifies to y=32x+2y = \frac{3}{2}x + 2y=23​x+2.This is our equation in **slope-intercept form**!

STEP 9

A. The equation of the perpendicular line in slope-intercept form is y=32x+2y = \frac{3}{2}x + 2y=23​x+2.

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1. Find the slope of the given line.
2. Find the slope of the perpendicular line.
3. Find the equation of the perpendicular line.

Now, we can easily see that the slope of the given line is $-\frac{2}{3}$ .
Awesome!

Remember, perpendicular lines have negative reciprocal slopes.
So, we need to flip the fraction and change the sign.
The negative reciprocal of $-\frac{2}{3}$ is $\frac{3}{2}$ .
This is the slope of our perpendicular line.

Let's simplify! $y + 1 = \frac{3}{2}(x + 2)$ .
Distribute the $\frac{3}{2}$ : $y + 1 = \frac{3}{2}x + \frac{3}{2} \cdot 2$ , which simplifies to $y + 1 = \frac{3}{2}x + 3$ .

Finally, subtract $1$ from both sides to get $y$ by itself: $y = \frac{3}{2}x + 3 - 1$ , which simplifies to $y = \frac{3}{2}x + 2$ .
This is our equation in slope-intercept form!

A. The equation of the perpendicular line in slope-intercept form is $y = \frac{3}{2}x + 2$ .