Math / Geometry

Question $2 x+6 y=10$
17. $y=-7$ $x=2$
18. $y=4 x-2$ $-x+4 y=0$ Write an equation in slope-intercept form of the line that passes through the grom point and is perpendicalar to the graph of the given equation.
19. $(0,0) ; y=-3 x+2$
20. $(-2,3): y=\frac{1}{2} x-1$
22. $(-3,2) ; x-2 y=7$
23. $(5,0) ; y+1=2(x-3)$
21. $(1,-2) ; y=5 x+4$
24. $(1,-6) ; x-2 y=4$
25. Urban Manning $\wedge$ path for a new city park will connect the park entrance to Main Strevet. The path should be perpendicular to Main Strevet.

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that's perpendicular to $y + 1 = 2(x - 3)$ and passes through the point $(5, 0)$ . Watch out! Remember, perpendicular lines have negative reciprocal slopes!
Don't forget to use the point-slope form to get the final equation.

STEP 2

1. Find the slope of the given line.
2. Find the perpendicular slope.
3. Use the point-slope form.
4. Simplify to slope-intercept form.

STEP 3

The given equation is $y + 1 = 2(x - 3)$ .
Let's rewrite it in slope-intercept form ( $y = mx + b$ ), which makes it easy to see the slope!
First, distribute the 2 on the right side: $y + 1 = 2 \cdot x - 2 \cdot 3$ , so $y + 1 = 2x - 6$ .
Now, subtract 1 from both sides to isolate $y$ : $y = 2x - 6 - 1$ , which simplifies to $y = 2x - 7$ .

STEP 4

Now, we can easily see that the slope of the given line is 2 (the coefficient of $x$ ).

STEP 5

To find the slope of a perpendicular line, we take the negative reciprocal of the original slope.
The negative reciprocal of 2 is $- \frac{1}{2}$ .
So, our perpendicular line has a slope of $- \frac{1}{2}$ .

STEP 6

We know our perpendicular line goes through the point $(5, 0)$ and has a slope of $- \frac{1}{2}$ .
We can use the point-slope form, $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is our point and $m$ is our slope.
Plugging in our values, we get $y - 0 = -\frac{1}{2}(x - 5)$ .

STEP 7

Let's simplify!
We have $y = -\frac{1}{2}(x - 5)$ .
Distributing the $- \frac{1}{2}$ , we get $y = -\frac{1}{2}x + \frac{5}{2}$ .
This is now in slope-intercept form!

STEP 8

The equation of the line perpendicular to $y + 1 = 2(x - 3)$ and passing through $(5, 0)$ is $y = -\frac{1}{2}x + \frac{5}{2}$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that's perpendicular to $y + 1 = 2(x - 3)$ and passes through the point $(5, 0)$ . Watch out! Remember, perpendicular lines have negative reciprocal slopes!
Don't forget to use the point-slope form to get the final equation.

STEP 2

1. Find the slope of the given line.
2. Find the perpendicular slope.
3. Use the point-slope form.
4. Simplify to slope-intercept form.

STEP 3

STEP 4

Now, we can easily see that the slope of the given line is 2 (the coefficient of $x$ ).

STEP 5

To find the slope of a perpendicular line, we take the negative reciprocal of the original slope.
The negative reciprocal of 2 is $- \frac{1}{2}$ .
So, our perpendicular line has a slope of $- \frac{1}{2}$ .

STEP 6

We know our perpendicular line goes through the point $(5, 0)$ and has a slope of $- \frac{1}{2}$ .
We can use the point-slope form, $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is our point and $m$ is our slope.
Plugging in our values, we get $y - 0 = -\frac{1}{2}(x - 5)$ .

STEP 7

Let's simplify!
We have $y = -\frac{1}{2}(x - 5)$ .
Distributing the $- \frac{1}{2}$ , we get $y = -\frac{1}{2}x + \frac{5}{2}$ .
This is now in slope-intercept form!

STEP 8

The equation of the line perpendicular to $y + 1 = 2(x - 3)$ and passing through $(5, 0)$ is $y = -\frac{1}{2}x + \frac{5}{2}$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the slope of the given line.2. Find the perpendicular slope.3. Use the point-slope form.4. Simplify to slope-intercept form.

STEP 3

STEP 4

Now, we can easily see that the slope of the given line is **2** (the coefficient of xxx).

STEP 5

To find the slope of a perpendicular line, we take the negative reciprocal of the original slope.The negative reciprocal of **2** is −12- \frac{1}{2}−21​.So, our perpendicular line has a slope of −12- \frac{1}{2}−21​.

STEP 6

STEP 7

Let's simplify!We have y=−12(x−5)y = -\frac{1}{2}(x - 5)y=−21​(x−5).Distributing the −12- \frac{1}{2}−21​, we get y=−12x+52y = -\frac{1}{2}x + \frac{5}{2}y=−21​x+25​.This is now in slope-intercept form!

STEP 8

The equation of the line perpendicular to y+1=2(x−3)y + 1 = 2(x - 3)y+1=2(x−3) and passing through (5,0)(5, 0)(5,0) is y=−12x+52y = -\frac{1}{2}x + \frac{5}{2}y=−21​x+25​.

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1. Find the slope of the given line.
2. Find the perpendicular slope.
3. Use the point-slope form.
4. Simplify to slope-intercept form.

Now, we can easily see that the slope of the given line is 2 (the coefficient of $x$ ).

To find the slope of a perpendicular line, we take the negative reciprocal of the original slope.
The negative reciprocal of 2 is $- \frac{1}{2}$ .
So, our perpendicular line has a slope of $- \frac{1}{2}$ .

Let's simplify!
We have $y = -\frac{1}{2}(x - 5)$ .
Distributing the $- \frac{1}{2}$ , we get $y = -\frac{1}{2}x + \frac{5}{2}$ .
This is now in slope-intercept form!

The equation of the line perpendicular to $y + 1 = 2(x - 3)$ and passing through $(5, 0)$ is $y = -\frac{1}{2}x + \frac{5}{2}$ .