Solved on Feb 27, 2024

Find the equation of the line passing through the point $(4,-3)$ and parallel to the line $y = -\frac{1}{6}x - 1$ .

STEP 1

Assumptions
1. The given line has the equation $y=-\frac{1}{6}x-1$ .
2. We need to find the equation of a line that is parallel to the given line and passes through the point $(4,-3)$ .

STEP 2

Recall that parallel lines have the same slope. Therefore, the slope of the line we are looking for will be the same as the slope of the given line.

STEP 3

Identify the slope of the given line by looking at its equation in slope-intercept form, which is $y=mx+b$ , where $m$ is the slope.
$Slope\, of\, given\, line = -\frac{1}{6}$

STEP 4

Since parallel lines have the same slope, the slope of the line we are looking for is also $-\frac{1}{6}$ .

STEP 5

Use the point-slope form of the equation of a line to find the equation of the line that passes through the point $(4,-3)$ with the slope $-\frac{1}{6}$ . 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.

STEP 6

Plug in the slope $-\frac{1}{6}$ and the point $(4,-3)$ into the point-slope form equation.
$y - (-3) = -\frac{1}{6}(x - 4)$

STEP 7

Simplify the equation by distributing the slope on the right side and moving the $-3$ to the left side.
$y + 3 = -\frac{1}{6}x + \frac{1}{6} \cdot 4$

STEP 8

Simplify the fraction on the right side.
$y + 3 = -\frac{1}{6}x + \frac{2}{3}$

STEP 9

Subtract 3 from both sides to get the equation in slope-intercept form $y=mx+b$ .
$y = -\frac{1}{6}x + \frac{2}{3} - 3$

STEP 10

Combine the constant terms on the right side.
$y = -\frac{1}{6}x + \frac{2}{3} - \frac{9}{3}$

STEP 11

Simplify the constant terms.
$y = -\frac{1}{6}x - \frac{7}{3}$
The equation of the line that is parallel to $y=-\frac{1}{6}x-1$ and passes through the point $(4,-3)$ is $y = -\frac{1}{6}x - \frac{7}{3}$ .

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Find the equation of the line passing through the point $(4,-3)$ and parallel to the line $y = -\frac{1}{6}x - 1$ .

STEP 1

Assumptions
1. The given line has the equation $y=-\frac{1}{6}x-1$ .
2. We need to find the equation of a line that is parallel to the given line and passes through the point $(4,-3)$ .

STEP 2

Recall that parallel lines have the same slope. Therefore, the slope of the line we are looking for will be the same as the slope of the given line.

STEP 3

Identify the slope of the given line by looking at its equation in slope-intercept form, which is $y=mx+b$ , where $m$ is the slope.
$Slope\, of\, given\, line = -\frac{1}{6}$

STEP 4

Since parallel lines have the same slope, the slope of the line we are looking for is also $-\frac{1}{6}$ .

STEP 5

Use the point-slope form of the equation of a line to find the equation of the line that passes through the point $(4,-3)$ with the slope $-\frac{1}{6}$ . 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.

STEP 6

Plug in the slope $-\frac{1}{6}$ and the point $(4,-3)$ into the point-slope form equation.
$y - (-3) = -\frac{1}{6}(x - 4)$

STEP 7

Simplify the equation by distributing the slope on the right side and moving the $-3$ to the left side.
$y + 3 = -\frac{1}{6}x + \frac{1}{6} \cdot 4$

STEP 8

Simplify the fraction on the right side.
$y + 3 = -\frac{1}{6}x + \frac{2}{3}$

STEP 9

Subtract 3 from both sides to get the equation in slope-intercept form $y=mx+b$ .
$y = -\frac{1}{6}x + \frac{2}{3} - 3$

STEP 10

Combine the constant terms on the right side.
$y = -\frac{1}{6}x + \frac{2}{3} - \frac{9}{3}$

STEP 11

Simplify the constant terms.
$y = -\frac{1}{6}x - \frac{7}{3}$
The equation of the line that is parallel to $y=-\frac{1}{6}x-1$ and passes through the point $(4,-3)$ is $y = -\frac{1}{6}x - \frac{7}{3}$ .

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Find the equation of the line passing through the point (4,−3)(4,-3)(4,−3) and parallel to the line y=−16x−1y = -\frac{1}{6}x - 1y=−61​x−1.

STEP 1

Assumptions1. The given line has the equation y=−16x−1y=-\frac{1}{6}x-1y=−61​x−1.2. We need to find the equation of a line that is parallel to the given line and passes through the point (4,−3)(4,-3)(4,−3).

STEP 2

Recall that parallel lines have the same slope. Therefore, the slope of the line we are looking for will be the same as the slope of the given line.

STEP 3

Identify the slope of the given line by looking at its equation in slope-intercept form, which is y=mx+by=mx+by=mx+b, where mmm is the slope.Slope of given line=−16Slope\, of\, given\, line = -\frac{1}{6}Slopeofgivenline=−61​

STEP 4

Since parallel lines have the same slope, the slope of the line we are looking for is also −16-\frac{1}{6}−61​.

STEP 5

STEP 6

Plug in the slope −16-\frac{1}{6}−61​ and the point (4,−3)(4,-3)(4,−3) into the point-slope form equation.y−(−3)=−16(x−4)y - (-3) = -\frac{1}{6}(x - 4)y−(−3)=−61​(x−4)

STEP 7

Simplify the equation by distributing the slope on the right side and moving the −3-3−3 to the left side.y+3=−16x+16⋅4y + 3 = -\frac{1}{6}x + \frac{1}{6} \cdot 4y+3=−61​x+61​⋅4

STEP 8

Simplify the fraction on the right side.y+3=−16x+23y + 3 = -\frac{1}{6}x + \frac{2}{3}y+3=−61​x+32​

STEP 9

Subtract 3 from both sides to get the equation in slope-intercept form y=mx+by=mx+by=mx+b.y=−16x+23−3y = -\frac{1}{6}x + \frac{2}{3} - 3y=−61​x+32​−3

STEP 10

Combine the constant terms on the right side.y=−16x+23−93y = -\frac{1}{6}x + \frac{2}{3} - \frac{9}{3}y=−61​x+32​−39​

STEP 11

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Find the equation of the line passing through the point $(4,-3)$ and parallel to the line $y = -\frac{1}{6}x - 1$ .

Assumptions
1. The given line has the equation $y=-\frac{1}{6}x-1$ .
2. We need to find the equation of a line that is parallel to the given line and passes through the point $(4,-3)$ .

Identify the slope of the given line by looking at its equation in slope-intercept form, which is $y=mx+b$ , where $m$ is the slope.
$Slope\, of\, given\, line = -\frac{1}{6}$

Since parallel lines have the same slope, the slope of the line we are looking for is also $-\frac{1}{6}$ .

Plug in the slope $-\frac{1}{6}$ and the point $(4,-3)$ into the point-slope form equation.
$y - (-3) = -\frac{1}{6}(x - 4)$

Simplify the equation by distributing the slope on the right side and moving the $-3$ to the left side.
$y + 3 = -\frac{1}{6}x + \frac{1}{6} \cdot 4$

Simplify the fraction on the right side.
$y + 3 = -\frac{1}{6}x + \frac{2}{3}$

Subtract 3 from both sides to get the equation in slope-intercept form $y=mx+b$ .
$y = -\frac{1}{6}x + \frac{2}{3} - 3$

Combine the constant terms on the right side.
$y = -\frac{1}{6}x + \frac{2}{3} - \frac{9}{3}$