What is the Point-Slope Formula of a Line?
The point-slope formula is a way to write the equation of a straight line when you know the slope of the line and one point it passes through. This form is especially helpful in algebra and coordinate geometry when you’re not given the y-intercept.
The point-slope formula is the following:
y − y₁ = m(x − x₁)
Where:
- (x₁, y₁) is a known point on the line.
- m is the slope of the line.
- (x, y) represents other points on the line that you can calculate.
The formula emphasizes how the slope (m) connects a known point on the line to any other point. You can easily rearrange it into slope-intercept form (y = mx + b) or standard form, depending on what you need.
Using the Point-Slope Formula
Suppose a line has a slope of 3 and passes through the point (2, 4). The point-slope formula is the following:
y − 4 = 3(x − 2)
This is the equation of the line in point-slope form. That graph below displays this line.
You can leave it like this and use it to calculate other points on the line. For example, if you substitute x = 4 into the point-slope equation:
y − 4 = 3(x − 2)
y − 4 = 3(4 − 2)
y − 4 = 6
y = 10
So, the point (4, 10) also lies on the line.
Or, you can simplify it into the slope-intercept form:
y − 4 = 3(x − 2)
y − 4 = 3x − 6
y = 3x − 2
The point-slope formula is a convenient tool for quickly writing the equation of a line when you’re not given the y-intercept but do know the slope and a point.
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