Solved on Dec 10, 2023

Find the equation of the line perpendicular to $5x-3y=18$ and passing through $(-9,10)$ .

STEP 1

Assumptions
1. We have a line with the equation $5x - 3y = 18$ .
2. We need to find the equation of a line that is perpendicular to this line.
3. The new line must pass through the point $(-9, 10)$ .

STEP 2

The slope of the given line can be found by rewriting the equation in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope.
$5x - 3y = 18$

STEP 3

Isolate $y$ on one side of the equation to find the slope of the given line.
$-3y = -5x + 18$

STEP 4

Divide both sides of the equation by $-3$ to solve for $y$ .
$y = \frac{5}{3}x - 6$

STEP 5

Now we can see that the slope of the given line is $\frac{5}{3}$ .

STEP 6

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, we take the negative reciprocal of $\frac{5}{3}$ to find the slope of the perpendicular line.
$m_{perpendicular} = -\frac{3}{5}$

STEP 7

Now we have the slope of the perpendicular line. We can use the point-slope form of the equation of a line to find the equation of the line that is perpendicular to the given line and passes through the point $(-9, 10)$ . The point-slope form is given by:
$y - y_1 = m(x - x_1)$
where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line.

STEP 8

Plug in the slope of the perpendicular line and the coordinates of the point $(-9, 10)$ into the point-slope form.
$y - 10 = -\frac{3}{5}(x - (-9))$

STEP 9

Simplify the equation by distributing the slope on the right side of the equation.
$y - 10 = -\frac{3}{5}x - \frac{3}{5} \cdot (-9)$

STEP 10

Multiply $-\frac{3}{5}$ by $-9$ to simplify further.
$y - 10 = -\frac{3}{5}x + \frac{27}{5}$

STEP 11

Convert the fraction $\frac{27}{5}$ to a decimal or leave as a fraction to make it easier to combine with $-10$ .
$y - 10 = -\frac{3}{5}x + 5.4$

STEP 12

Now, add $10$ to both sides of the equation to solve for $y$ .
$y = -\frac{3}{5}x + 5.4 + 10$

STEP 13

Combine the constant terms on the right side of the equation.
$y = -\frac{3}{5}x + 15.4$
The equation of the line that is perpendicular to $5x - 3y = 18$ and passes through the point $(-9, 10)$ is:
$y = -\frac{3}{5}x + 15.4$

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Find the equation of the line perpendicular to $5x-3y=18$ and passing through $(-9,10)$ .

STEP 1

Assumptions
1. We have a line with the equation $5x - 3y = 18$ .
2. We need to find the equation of a line that is perpendicular to this line.
3. The new line must pass through the point $(-9, 10)$ .

STEP 2

The slope of the given line can be found by rewriting the equation in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope.
$5x - 3y = 18$

STEP 3

Isolate $y$ on one side of the equation to find the slope of the given line.
$-3y = -5x + 18$

STEP 4

Divide both sides of the equation by $-3$ to solve for $y$ .
$y = \frac{5}{3}x - 6$

STEP 5

Now we can see that the slope of the given line is $\frac{5}{3}$ .

STEP 6

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, we take the negative reciprocal of $\frac{5}{3}$ to find the slope of the perpendicular line.
$m_{perpendicular} = -\frac{3}{5}$

STEP 7

STEP 8

Plug in the slope of the perpendicular line and the coordinates of the point $(-9, 10)$ into the point-slope form.
$y - 10 = -\frac{3}{5}(x - (-9))$

STEP 9

Simplify the equation by distributing the slope on the right side of the equation.
$y - 10 = -\frac{3}{5}x - \frac{3}{5} \cdot (-9)$

STEP 10

Multiply $-\frac{3}{5}$ by $-9$ to simplify further.
$y - 10 = -\frac{3}{5}x + \frac{27}{5}$

STEP 11

Convert the fraction $\frac{27}{5}$ to a decimal or leave as a fraction to make it easier to combine with $-10$ .
$y - 10 = -\frac{3}{5}x + 5.4$

STEP 12

Now, add $10$ to both sides of the equation to solve for $y$ .
$y = -\frac{3}{5}x + 5.4 + 10$

STEP 13

Combine the constant terms on the right side of the equation.
$y = -\frac{3}{5}x + 15.4$
The equation of the line that is perpendicular to $5x - 3y = 18$ and passes through the point $(-9, 10)$ is:
$y = -\frac{3}{5}x + 15.4$

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Find the equation of the line perpendicular to 5x−3y=185x-3y=185x−3y=18 and passing through (−9,10)(-9,10)(−9,10).

STEP 1

Assumptions1. We have a line with the equation 5x−3y=185x - 3y = 185x−3y=18.2. We need to find the equation of a line that is perpendicular to this line.3. The new line must pass through the point (−9,10)(-9, 10)(−9,10).

STEP 2

The slope of the given line can be found by rewriting the equation in slope-intercept form, which is y=mx+by = mx + by=mx+b, where mmm is the slope.5x−3y=185x - 3y = 185x−3y=18

STEP 3

Isolate yyy on one side of the equation to find the slope of the given line.−3y=−5x+18-3y = -5x + 18−3y=−5x+18

STEP 4

Divide both sides of the equation by −3-3−3 to solve for yyy.y=53x−6y = \frac{5}{3}x - 6y=35​x−6

STEP 5

Now we can see that the slope of the given line is 53\frac{5}{3}35​.

STEP 6

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, we take the negative reciprocal of 53\frac{5}{3}35​ to find the slope of the perpendicular line.mperpendicular=−35m_{perpendicular} = -\frac{3}{5}mperpendicular​=−53​

STEP 7

STEP 8

Plug in the slope of the perpendicular line and the coordinates of the point (−9,10)(-9, 10)(−9,10) into the point-slope form.y−10=−35(x−(−9))y - 10 = -\frac{3}{5}(x - (-9))y−10=−53​(x−(−9))

STEP 9

Simplify the equation by distributing the slope on the right side of the equation.y−10=−35x−35⋅(−9)y - 10 = -\frac{3}{5}x - \frac{3}{5} \cdot (-9)y−10=−53​x−53​⋅(−9)

STEP 10

Multiply −35-\frac{3}{5}−53​ by −9-9−9 to simplify further.y−10=−35x+275y - 10 = -\frac{3}{5}x + \frac{27}{5}y−10=−53​x+527​

STEP 11

Convert the fraction 275\frac{27}{5}527​ to a decimal or leave as a fraction to make it easier to combine with −10-10−10.y−10=−35x+5.4y - 10 = -\frac{3}{5}x + 5.4y−10=−53​x+5.4

STEP 12

Now, add 101010 to both sides of the equation to solve for yyy.y=−35x+5.4+10y = -\frac{3}{5}x + 5.4 + 10y=−53​x+5.4+10

STEP 13

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Find the equation of the line perpendicular to $5x-3y=18$ and passing through $(-9,10)$ .

Assumptions
1. We have a line with the equation $5x - 3y = 18$ .
2. We need to find the equation of a line that is perpendicular to this line.
3. The new line must pass through the point $(-9, 10)$ .

The slope of the given line can be found by rewriting the equation in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope.
$5x - 3y = 18$

Isolate $y$ on one side of the equation to find the slope of the given line.
$-3y = -5x + 18$

Divide both sides of the equation by $-3$ to solve for $y$ .
$y = \frac{5}{3}x - 6$

Now we can see that the slope of the given line is $\frac{5}{3}$ .

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, we take the negative reciprocal of $\frac{5}{3}$ to find the slope of the perpendicular line.
$m_{perpendicular} = -\frac{3}{5}$

Plug in the slope of the perpendicular line and the coordinates of the point $(-9, 10)$ into the point-slope form.
$y - 10 = -\frac{3}{5}(x - (-9))$

Simplify the equation by distributing the slope on the right side of the equation.
$y - 10 = -\frac{3}{5}x - \frac{3}{5} \cdot (-9)$

Multiply $-\frac{3}{5}$ by $-9$ to simplify further.
$y - 10 = -\frac{3}{5}x + \frac{27}{5}$

Convert the fraction $\frac{27}{5}$ to a decimal or leave as a fraction to make it easier to combine with $-10$ .
$y - 10 = -\frac{3}{5}x + 5.4$

Now, add $10$ to both sides of the equation to solve for $y$ .
$y = -\frac{3}{5}x + 5.4 + 10$