Point Slope Form
The point-slope formula is \(y-y_1=m(x-x_1)\) where \(m\) is the slope and \((x_1,y_1)\) is a point on the line.
Point-slope form is a quick way to write the equation of a line when you know the slope and one point on the line. It’s especially helpful when the point is not the y-intercept, or when the point includes negatives and you want to keep your equation accurate without extra steps.
How It Works
Identify the slope
Find the slope of the line, m. If the slope is given, use that value. If you are given 2 points, calculate the slope using the slope formula.
$$m=\frac{y_2-y_1}{x_2-x_1}$$
Identify the point \((x_1,y_1)\)
Choose a point the line passes through. If one point is given, use it. If two points are given, you may use either point as \((x_1,y_1)\). An example is shown below.
$$(x_1,y_1)=(2,1)$$
Substitute into point-slope form
Plug the slope and point into the point-slope formula: \(y-y_1=m(x-x_1)\). An example is shown using the point above.
$$y-1=m(x−2)$$
Try These Examples
Write an equation of the line in point-slope form \(y-y_1=m(x-x_1)\) slope = 4, point = (2, -1). $$y−(−1)=4(x−2)$$ $$y+1=4(x−2)$$
Be extra careful when the point has negative coordinates. Subtracting a negative becomes addition. Keep parentheses until you simplify. Write an equation of the line in point-slope form \(y-y_1=m(x-x_1)\) slope \(=\frac{1}{2} \), point = \( (-4,\,8) \) $$y−8=\frac{1}{2}(x−(−4))$$ Write an equation of the line in point-slope form \(y-y_1=m(x-x_1)\) slope \(=\frac{4}{3} \), point = \( (1,\,0) \) $$y−0=\frac{4}{3}(x-1)$$
If the point has a y-value of 0, you can drop the – 0 in the equation as shown above. The same thing works for an x-value of 0.
$$y−8=\frac{1}{2}(x+4)$$
$$y=\frac{4}{3}(x-1)$$
Watch Out
Common Mistake
Mixing up \(x_1\) and \(y_1\). Make sure you substitute the numbers in the correct spot in the formula.
Mixing up \(x_1\) and \(y_1\). Make sure you substitute the numbers in the correct spot in the formula.
Keep Practicing
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