Point-Slope Form

When you know the slope of a line and one point it passes through — but not necessarily the y-intercept — the point-slope form is the fastest way to write the equation. It is especially useful in applied settings where you have a measured rate of change and a single data point.

The Point-Slope Formula

Given a line with slope $m$ that passes through the point $(x_1, y_1)$, the equation of the line is:

$y - y_1 = m(x - x_1)$

This formula comes directly from the slope definition. If $(x, y)$ is any point on the line and $(x_1, y_1)$ is the known point, then:

$m = \frac{y - y_1}{x - x_1}$

Multiply both sides by $(x - x_1)$ and you get the point-slope form.

When to Use Point-Slope Form

Use point-slope form when you are given:

• A slope and a point (that is not the y-intercept)
• Two points (calculate the slope first, then use either point)
• A real-world rate of change and one known measurement

If you already know the y-intercept, slope-intercept form ($y = mx + b$) may be more direct. But point-slope form works in every situation — including when the y-intercept is unknown or inconvenient to find.

Writing Equations from a Point and Slope

Example 1: Basic point-slope equation

Write the equation of a line with slope $4$ passing through $(3, 7)$.

Step 1 — Identify the values: $m = 4$, $x_1 = 3$, $y_1 = 7$.

Step 2 — Substitute into the formula:

$y - 7 = 4(x - 3)$

That is a complete equation of the line in point-slope form.

Example 2: Negative slope

Write the equation of a line with slope $-\frac{2}{3}$ passing through $(6, 1)$.

Step 1 — Identify the values: $m = -\frac{2}{3}$, $x_1 = 6$, $y_1 = 1$.

Step 2 — Substitute:

$y - 1 = -\frac{2}{3}(x - 6)$

Example 3: Point with negative coordinates

Write the equation of a line with slope $5$ passing through $(-2, -4)$.

Step 1 — Identify: $m = 5$, $x_1 = -2$, $y_1 = -4$.

Step 2 — Substitute carefully (watch the double negatives):

$y - (-4) = 5(x - (-2))$

$y + 4 = 5(x + 2)$

Writing an Equation from Two Points

When you are given two points, first calculate the slope, then use either point in the point-slope formula.

Example 4: Two points

Write the equation of the line through $(2, 5)$ and $(8, 17)$.

Step 1 — Find the slope:

$m = \frac{17 - 5}{8 - 2} = \frac{12}{6} = 2$

Step 2 — Choose either point. Using $(2, 5)$:

$y - 5 = 2(x - 2)$

You would get the same line if you used $(8, 17)$ instead:

$y - 17 = 2(x - 8)$

Both equations describe the same line. You can verify by converting either to slope-intercept form — both simplify to $y = 2x + 1$.

Converting Point-Slope to Slope-Intercept Form

To graph a line or compare equations, you often want slope-intercept form. Distribute and simplify.

Example 5: Full conversion

Convert $y - 7 = 4(x - 3)$ to slope-intercept form.

Step 1 — Distribute the slope:

$y - 7 = 4x - 12$

Step 2 — Add $7$ to both sides:

$y = 4x - 12 + 7$

$y = 4x - 5$

So the slope is $4$ and the y-intercept is $-5$.

Example 6: Conversion with fractions

Convert $y - 1 = -\frac{2}{3}(x - 6)$ to slope-intercept form.

Step 1 — Distribute:

$y - 1 = -\frac{2}{3}x + \frac{2}{3} \cdot 6$

$y - 1 = -\frac{2}{3}x + 4$

Step 2 — Add $1$:

$y = -\frac{2}{3}x + 5$

Converting Point-Slope to Standard Form

Sometimes you need the equation in standard form ($Ax + By = C$). Start from point-slope, distribute, then rearrange so both variable terms are on one side.

Example 7: Convert to standard form

Convert $y + 4 = 5(x + 2)$ to standard form.

Step 1 — Distribute:

$y + 4 = 5x + 10$

Step 2 — Move all variable terms to one side and constants to the other:

$y = 5x + 6$

$-5x + y = 6$

Multiply by $-1$ so the $x$-coefficient is positive (standard convention):

$5x - y = -6$

Real-World Application: Carpentry — Calculating Stair Dimensions

A carpenter is building stairs. Building code requires a specific rise-to-run ratio. After measuring the site, the carpenter knows:

• The staircase has a slope (rise/run) of $\frac{7}{10}$ (each step rises 7 inches over a 10-inch tread)
• The bottom of the first step is at the point $(0, 0)$ (ground level)
• The landing at the top is at the point $(100, 70)$ — 100 inches of horizontal run and 70 inches of total rise

Using any intermediate point — say the top of the third step at $(30, 21)$ — and the known slope, the carpenter writes:

$y - 21 = \frac{7}{10}(x - 30)$

This equation lets the carpenter calculate the height at any horizontal position along the staircase. For example, at $x = 50$ inches:

$y - 21 = \frac{7}{10}(50 - 30) = \frac{7}{10}(20) = 14$

$y = 35 \text{ inches}$

The staircase is 35 inches high at 50 inches of horizontal distance. The carpenter uses these calculations to cut stringers (the diagonal support boards) to exact length and verify each step meets the code-required dimensions.

Choosing Between Forms

Common Mistakes to Avoid

1. Flipping the signs in the formula. The formula is $y - y_1 = m(x - x_1)$, with subtraction. If your point is $(3, 7)$, write $y - 7 = m(x - 3)$, not $y + 7$ or $x + 3$.

2. Double-negative errors. When the point has negative coordinates, be very careful: $y - (-4)$ becomes $y + 4$, and $x - (-2)$ becomes $x + 2$.

3. Forgetting to distribute the slope to both terms. When converting $y - 7 = 4(x - 3)$, distribute the $4$ to get $4x - 12$, not $4x - 3$.

4. Using the wrong point. When given two points, you can use either one — but make sure you use the slope you calculated from both points, not a number from just one point.

5. Leaving the equation unsimplified when simplification is requested. Point-slope form is a valid final answer, but if a problem asks for slope-intercept or standard form, you must convert completely.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Point-slope form is $y - y_1 = m(x - x_1)$ — use it when you know a slope and a point
• It is the most flexible form for writing equations because it does not require the y-intercept
• To convert to slope-intercept form, distribute the slope and solve for $y$
• When given two points, calculate the slope first, then use either point in the formula
• In applied problems, the slope is the rate and the point is a known measurement

Return to Algebra 1 for more topics in this section.