Math / Algebra

Question16. Write the equation in slope-intercept form of the line that passes through the points $(-6,-10)$ and $(21,8)$ .
17. What is the equation in slope-intercept form of a line that passes through the point $(2,4)$ and is perpendicular to the x-axis? A. $y=2$ C. $y=4$ B. $x=2$ D. $x=4$
18. What is the equation in slope-intercept form of a line that passes through the point $(8,-14)$ and is parallel to $y=\frac{1}{2} x-4 ?$ A. $y=-2 x+2$ C. $y=-2 x-4$ B. $y=\frac{1}{2} x-18$ D. $y=\frac{1}{2} x+2$
19. Write an equation for the line that is perpendicular to the $y$ -axis and has the same $y$ -intercept as the line $y=3.6 x-4.9$ . A. $y=-4.9$ C. $y=3.6$ B. $x=3.6$ D. $x=-4.9$
20. What is the equation of the line that is parallel to $-4 x-2 y=12$ and passes through the point $(4,1) ?$ A. $y=-2 x-7$ B. $y=2 x-7$ C. $y=2 x-9$ D. $y=\frac{1}{2} x-1$

Studdy Solution

STEP 1

What is this asking? We've got a bunch of line equation problems!
We need to find equations of lines given different conditions, like passing through certain points or being parallel/perpendicular to other lines. Watch out! Remember the slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
Don't mix these up!
Also, parallel lines have the same slope, and perpendicular lines have negative reciprocal slopes.

STEP 2

1. Find Equation Given Two Points
2. Find Equation of Line Perpendicular to x-axis
3. Find Equation of Line Parallel to Given Line
4. Find Equation of Line Perpendicular to y-axis
5. Find Equation of Parallel Line and Given Point

STEP 3

We're given two points, $(-6, -10)$ and $(21, 8)$ .
Let's find the slope!
The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Plugging in our points, we get $m = \frac{8 - (-10)}{21 - (-6)} = \frac{18}{27} = \frac{2}{3}$ .
Our slope is $\frac{2}{3}$ .

STEP 4

Now, we use the point-slope form: $y - y_1 = m(x - x_1)$ .
Let's use the point $(-6, -10)$ and our slope $\frac{2}{3}$ .
We get $y - (-10) = \frac{2}{3}(x - (-6))$ , which simplifies to $y + 10 = \frac{2}{3}(x + 6)$ .

STEP 5

Finally, let's get this into slope-intercept form.
Distribute the $\frac{2}{3}$ : $y + 10 = \frac{2}{3}x + 4$ .
Subtract 10 from both sides: $y = \frac{2}{3}x - 6$ .
Boom!

STEP 6

A line perpendicular to the x-axis is a vertical line.
Vertical lines have the equation $x = c$ , where $c$ is a constant.
Since our line goes through $(2, 4)$ , the equation must be $x = 2$ .

STEP 7

Our line is parallel to $y = \frac{1}{2}x - 4$ , so it has the same slope, which is $\frac{1}{2}$ .

STEP 8

We know the line passes through $(8, -14)$ .
Using point-slope form, we get $y - (-14) = \frac{1}{2}(x - 8)$ , which simplifies to $y + 14 = \frac{1}{2}x - 4$ .

STEP 9

Subtract 14 from both sides: $y = \frac{1}{2}x - 18$ .
Perfect!

STEP 10

A line perpendicular to the y-axis is a horizontal line, which has the form $y = c$ .

STEP 11

The y-intercept of $y = 3.6x - 4.9$ is $-4.9$ .
Since our line has the same y-intercept, its equation is $y = -4.9$ .

STEP 12

First, let's rewrite $-4x - 2y = 12$ in slope-intercept form.
Add $4x$ to both sides: $-2y = 4x + 12$ .
Divide by $-2$ : $y = -2x - 6$ .

STEP 13

The slope of this line is $-2$ .
Since our line is parallel, it also has a slope of $-2$ .

STEP 14

Our line passes through $(4, 1)$ .
Using point-slope form, we have $y - 1 = -2(x - 4)$ , which simplifies to $y - 1 = -2x + 8$ .

STEP 15

Add 1 to both sides: $y = -2x + 9$ .
Done!

STEP 16

16. $y = \frac{2}{3}x - 6$
17. B. $x = 2$
18. B. $y = \frac{1}{2}x - 18$
19. A. $y = -4.9$
20. No correct option provided.
Correct answer is $y = -2x + 9$

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Studdy Solution

STEP 1

STEP 2

1. Find Equation Given Two Points
2. Find Equation of Line Perpendicular to x-axis
3. Find Equation of Line Parallel to Given Line
4. Find Equation of Line Perpendicular to y-axis
5. Find Equation of Parallel Line and Given Point

STEP 3

We're given two points, $(-6, -10)$ and $(21, 8)$ .
Let's find the slope!
The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Plugging in our points, we get $m = \frac{8 - (-10)}{21 - (-6)} = \frac{18}{27} = \frac{2}{3}$ .
Our slope is $\frac{2}{3}$ .

STEP 4

Now, we use the point-slope form: $y - y_1 = m(x - x_1)$ .
Let's use the point $(-6, -10)$ and our slope $\frac{2}{3}$ .
We get $y - (-10) = \frac{2}{3}(x - (-6))$ , which simplifies to $y + 10 = \frac{2}{3}(x + 6)$ .

STEP 5

Finally, let's get this into slope-intercept form.
Distribute the $\frac{2}{3}$ : $y + 10 = \frac{2}{3}x + 4$ .
Subtract 10 from both sides: $y = \frac{2}{3}x - 6$ .
Boom!

STEP 6

A line perpendicular to the x-axis is a vertical line.
Vertical lines have the equation $x = c$ , where $c$ is a constant.
Since our line goes through $(2, 4)$ , the equation must be $x = 2$ .

STEP 7

Our line is parallel to $y = \frac{1}{2}x - 4$ , so it has the same slope, which is $\frac{1}{2}$ .

STEP 8

We know the line passes through $(8, -14)$ .
Using point-slope form, we get $y - (-14) = \frac{1}{2}(x - 8)$ , which simplifies to $y + 14 = \frac{1}{2}x - 4$ .

STEP 9

Subtract 14 from both sides: $y = \frac{1}{2}x - 18$ .
Perfect!

STEP 10

A line perpendicular to the y-axis is a horizontal line, which has the form $y = c$ .

STEP 11

The y-intercept of $y = 3.6x - 4.9$ is $-4.9$ .
Since our line has the same y-intercept, its equation is $y = -4.9$ .

STEP 12

First, let's rewrite $-4x - 2y = 12$ in slope-intercept form.
Add $4x$ to both sides: $-2y = 4x + 12$ .
Divide by $-2$ : $y = -2x - 6$ .

STEP 13

The slope of this line is $-2$ .
Since our line is parallel, it also has a slope of $-2$ .

STEP 14

Our line passes through $(4, 1)$ .
Using point-slope form, we have $y - 1 = -2(x - 4)$ , which simplifies to $y - 1 = -2x + 8$ .

STEP 15

Add 1 to both sides: $y = -2x + 9$ .
Done!

STEP 16

16. $y = \frac{2}{3}x - 6$
17. B. $x = 2$
18. B. $y = \frac{1}{2}x - 18$
19. A. $y = -4.9$
20. No correct option provided.
Correct answer is $y = -2x + 9$

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Studdy Solution

STEP 1

STEP 2

1. Find Equation Given Two Points2. Find Equation of Line Perpendicular to x-axis3. Find Equation of Line Parallel to Given Line4. Find Equation of Line Perpendicular to y-axis5. Find Equation of Parallel Line and Given Point

STEP 3

STEP 4

STEP 5

Finally, let's get this into **slope-intercept form**.Distribute the 23\frac{2}{3}32​: y+10=23x+4y + 10 = \frac{2}{3}x + 4y+10=32​x+4.Subtract 10 from both sides: y=23x−6y = \frac{2}{3}x - 6y=32​x−6.Boom!

STEP 6

A line perpendicular to the x-axis is a *vertical* line.Vertical lines have the equation x=cx = cx=c, where ccc is a constant.Since our line goes through (2,4)(2, 4)(2,4), the equation must be x=2x = 2x=2.

STEP 7

Our line is parallel to y=12x−4y = \frac{1}{2}x - 4y=21​x−4, so it has the **same slope**, which is 12\frac{1}{2}21​.

STEP 8

We know the line passes through (8,−14)(8, -14)(8,−14).Using point-slope form, we get y−(−14)=12(x−8)y - (-14) = \frac{1}{2}(x - 8)y−(−14)=21​(x−8), which simplifies to y+14=12x−4y + 14 = \frac{1}{2}x - 4y+14=21​x−4.

STEP 9

Subtract 14 from both sides: y=12x−18y = \frac{1}{2}x - 18y=21​x−18.Perfect!

STEP 10

A line perpendicular to the y-axis is a *horizontal* line, which has the form y=cy = cy=c.

STEP 11

The y-intercept of y=3.6x−4.9y = 3.6x - 4.9y=3.6x−4.9 is −4.9-4.9−4.9.Since our line has the same y-intercept, its equation is y=−4.9y = -4.9y=−4.9.

STEP 12

First, let's rewrite −4x−2y=12-4x - 2y = 12−4x−2y=12 in slope-intercept form.Add 4x4x4x to both sides: −2y=4x+12-2y = 4x + 12−2y=4x+12.Divide by −2-2−2: y=−2x−6y = -2x - 6y=−2x−6.

STEP 13

The **slope** of this line is −2-2−2.Since our line is parallel, it also has a slope of −2-2−2.

STEP 14

Our line passes through (4,1)(4, 1)(4,1).Using point-slope form, we have y−1=−2(x−4)y - 1 = -2(x - 4)y−1=−2(x−4), which simplifies to y−1=−2x+8y - 1 = -2x + 8y−1=−2x+8.

STEP 15

Add 1 to both sides: y=−2x+9y = -2x + 9y=−2x+9.Done!

STEP 16

16. y=23x−6y = \frac{2}{3}x - 6y=32​x−617. B. x=2x = 2x=218. B. y=12x−18y = \frac{1}{2}x - 18y=21​x−1819. A. y=−4.9y = -4.9y=−4.920. No correct option provided.Correct answer is y=−2x+9y = -2x + 9y=−2x+9

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1. Find Equation Given Two Points
2. Find Equation of Line Perpendicular to x-axis
3. Find Equation of Line Parallel to Given Line
4. Find Equation of Line Perpendicular to y-axis
5. Find Equation of Parallel Line and Given Point

Finally, let's get this into slope-intercept form.
Distribute the $\frac{2}{3}$ : $y + 10 = \frac{2}{3}x + 4$ .
Subtract 10 from both sides: $y = \frac{2}{3}x - 6$ .
Boom!

A line perpendicular to the x-axis is a vertical line.
Vertical lines have the equation $x = c$ , where $c$ is a constant.
Since our line goes through $(2, 4)$ , the equation must be $x = 2$ .

Our line is parallel to $y = \frac{1}{2}x - 4$ , so it has the same slope, which is $\frac{1}{2}$ .

We know the line passes through $(8, -14)$ .
Using point-slope form, we get $y - (-14) = \frac{1}{2}(x - 8)$ , which simplifies to $y + 14 = \frac{1}{2}x - 4$ .

Subtract 14 from both sides: $y = \frac{1}{2}x - 18$ .
Perfect!

A line perpendicular to the y-axis is a horizontal line, which has the form $y = c$ .

The y-intercept of $y = 3.6x - 4.9$ is $-4.9$ .
Since our line has the same y-intercept, its equation is $y = -4.9$ .

First, let's rewrite $-4x - 2y = 12$ in slope-intercept form.
Add $4x$ to both sides: $-2y = 4x + 12$ .
Divide by $-2$ : $y = -2x - 6$ .

The slope of this line is $-2$ .
Since our line is parallel, it also has a slope of $-2$ .

Our line passes through $(4, 1)$ .
Using point-slope form, we have $y - 1 = -2(x - 4)$ , which simplifies to $y - 1 = -2x + 8$ .

Add 1 to both sides: $y = -2x + 9$ .
Done!

16. $y = \frac{2}{3}x - 6$
17. B. $x = 2$
18. B. $y = \frac{1}{2}x - 18$
19. A. $y = -4.9$
20. No correct option provided.
Correct answer is $y = -2x + 9$