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Point-Slope Formula
Finding the equation of a line is a fundamental task in algebra. The point-slope formula is the easiest way to write linear equations when you know the rate of change and a single coordinate. For quick conversions, use our point-slope form calculator or check slope variables with the slope calculator.
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What is the point-slope formula?
The point-slope equation is derived directly from the definition of a line's slope (rise over run). If you multiply both sides of the slope formula m = (y - y₁) / (x - x₁) by the denominator (x - x₁), you get the point-slope equation:
y − y₁ = m(x − x₁)
Variables Explained
- m = Slope of the line (representing steepness and direction).
- (x₁, y₁) = Coordinates of a known point on the line.
- x and y = Variables that remain in the final equation representing any general coordinate (x, y) on the line.
Worked Examples
Example 1: Given a Point and Slope
Write the equation of the line that passes through the point (2, 5) with a slope of 3. Convert it to slope-intercept form.
- Formula: y − y₁ = m(x − x₁)
- Given: x₁ = 2, y₁ = 5, m = 3
- Step 1 (Sub): y − 5 = 3(x − 2)
- Step 2 (Distribute): y − 5 = 3x − 6
- Step 3 (Isolate y): y = 3x − 6 + 5
- Result: y = 3x − 1
Example 2: Given Two Points
Write the equation of the line passing through (1, 3) and (3, 7).
- Step 1 (Find Slope): m = (7 − 3) ÷ (3 − 1) = 4 ÷ 2 = 2
- Step 2 (Apply Point-Slope with point (1,3)): y − 3 = 2(x − 1)
- Step 3 (Convert): y − 3 = 2x − 2
- Result: y = 2x + 1
Common Mistakes with Point-Slope
- Sign Errors with Negative Coordinates: If x₁ or y₁ is negative, subtracting a negative makes a positive. For example, if your point is (-3, -4), the formula becomes y - (-4) = m(x - (-3)), which simplifies to y + 4 = m(x + 3).
- Mixing up X and Y: Be careful not to plug the x-coordinate into y₁ and vice versa. Always check that y₁ is the vertical value and x₁ is the horizontal value.
- Failing to Distribute the Slope: When simplifying m(x - x₁), remember to multiply m by **both** x and x₁. A common error is writing 2(x - 4) = 2x - 4 instead of 2x - 8.
Run the numbers
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Frequently asked questions
The point-slope formula is y − y₁ = m(x − x₁), where m is the slope of the line and (x₁, y₁) represents a known coordinate point on the line.
Point-slope form is y − y₁ = m(x − x₁) and is best when starting with a slope and any random point. Slope-intercept form is y = mx + b, which requires knowing the specific y-intercept (b).
Distribute the slope (m) to the terms inside the parentheses, then isolate y by adding y₁ to both sides. For example, y − 3 = 2(x − 1) simplifies to y − 3 = 2x − 2, which becomes y = 2x + 1.
No. Vertical lines have an undefined slope (m). Their equations are written in the form x = k, where k is the x-coordinate of all points on the line.