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# Point-Slope Form for the Equation of a Line

Suppose we want to find the equation of the line that passes through the point (2, 5) with slope .

 Recall the formula for the slope of a line through the points (x1, y1) and (x2, y2). We know the slope, m, is . m We are given the coordinates of one point, (2, 5). So we substitute 2 for x1 and 5 for y1. We use (x, y) to represent another point on the line. Substitute x for x2 and y for y2. To clear the fraction on the right side, multiply both sides by x - 2. = y - 5 Finally, exchange the left and right sides of the equation. y - 5
This equation of the line is in point-slope form.

It displays the coordinates of a point, (2, 5), and the slope, .

We can easily write the equation of a line in point-slope form when we are given the coordinates of a point on the line and the slope of the line.

Definition â€” Point-Slope Form for the Equation of a Line

The point-slope form for the equation of a line that passes through the point (x1, y1) with slope m is:

y - y1 = m(x - x1)

Note that m, x1, and y1 are constants, whereas x and y are variables.

Example

Find the equation of the line that passes through the point (7, -4) with slope 3.

Write your answer in point-slope form.

Solution

 Here is the point-slope form for the equation of a line.The slope, m, is 3. So, replace m with 3. A point, (x1, y1), on the line is (7, -4). So, replace x1 with 7, and replace y1 with -4. y - y1 y - y1 y - (-4) = m(x - x1) = 3(x - x1) = 3(x - 7)
The equation of the line in point-slope form is y - (-4) = 3(x - 7).

Note:

You can simplify the left side of the equation to obtain y + 4 = 3(x - 7).

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