Start free
  • Analytic Geometry: Lines and Circles

How do you find the equation of a line from a point and a slope?

Quick answer

Use point-slope form: y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of the known point. Substitute the values of m, x1, and y1 into the formula, distribute m across the parentheses, and isolate y to write the final equation in slope-intercept form, y = mx + b.

The topic explained

The point-slope formula

Every straight line has a constant slope, written as m, which represents its rate of change. When you know that slope and any single point (x1, y1) that sits on the line, you can pinpoint the line's exact equation on the coordinate grid.

To write the equation of a line from a point and slope, use the point-slope formula: y - y1 = m(x - x1). This formula comes straight from the definition of slope as rise over run: m = (y - y1) / (x - x1). Multiplying both sides by (x - x1) gives the standard point-slope equation.

Step-by-step method

First, identify your given values: slope m and coordinates x1 and y1. Second, substitute these numbers into y - y1 = m(x - x1). Watch your signs carefully when plugging in negative values for x1 or y1.

Third, multiply m through the parentheses. Finally, add or subtract y1 to isolate y on the left side. This leaves the line in slope-intercept form, y = mx + b, where b is the y-intercept.

A concrete example with negative numbers

Suppose a line passes through point P(-2, 5) with a slope of 3. Here, m = 3, x1 = -2, and y1 = 5. Substitute these into the formula: y - 5 = 3(x - (-2)).

Simplify the double negative inside the parentheses to get y - 5 = 3(x + 2). Distribute the 3: y - 5 = 3x + 6. Now add 5 to both sides: y = 3x + 11. The line has slope 3 and crosses the y-axis at (0, 11).

Common pitfalls to watch for

The most frequent error is swapping x1 and y1 during substitution. Always check that the x-coordinate goes inside the parentheses with x, and the y-coordinate goes on the opposite side with y.

A second common slip is distributing the slope incorrectly across negative values. If x1 is -4, x - x1 becomes (x + 4). Distributing a negative slope like -2 across (x + 4) yields -2x - 8, not -2x + 8.

Free printable worksheet

Finding the Equation of a Line: Point and Slope Practice

For 11th-grade analytic geometry students. Write the equation of each line in slope-intercept form (y = mx + b) using the given point and slope.

Print the worksheet Regenerate new worksheet

Free, no sign-up. The answer key prints on the second page. Regenerating a new worksheet on the same topic needs a free account.

Free printable worksheetlessoons.com
Finding the Equation of a Line: Point and Slope PracticeName:

  1. Point: (2, 3), Slope: 4

  2. Point: (5, 1), Slope: -2

  3. Point: (-3, 4), Slope: 2

  4. Point: (-4, -6), Slope: -3

  5. Point: (6, -2), Slope: 1/2

  6. Point: (-1, 7), Slope: -2/3

  7. A line passes through (4, -5) with slope 0. Find its equation.

  8. A line passes through (-3, 8) with an undefined slope. Find its equation.

More topics and worksheets at lessoons.com© 2026 Lessons

Common questions

Can I use y = mx + b directly instead of point-slope form?

Yes. Plug your known slope m and the coordinates x and y into y = mx + b. Solve the resulting equation for b. Once you have b, write the final equation using your numerical values for m and b. Both approaches produce the exact same equation.

What if the slope of the line is zero?

A slope of zero means the line is completely horizontal. Substituting m = 0 into y - y1 = m(x - x1) leaves y - y1 = 0, which simplifies to y = y1. For example, through (3, 7) with slope 0, the equation is simply y = 7.

What if the slope is undefined?

An undefined slope indicates a vertical line. Vertical lines cannot be written in y = mx + b form because they have no defined slope and do not cross the y-axis (unless on the axis). Instead, their equation is always x = a, where a is the x-coordinate of the point.

How do I find the equation if I am given two points instead of a slope?

First, calculate the slope m using the formula m = (y2 - y1) / (x2 - x1). Once you have m, pick either of the two points and use the standard point-slope formula y - y1 = m(x - x1) to finish writing the equation.

How does Lessi help with coordinate geometry?

Lessi is an AI tutor that explains analytic geometry concepts step by step and checks your line equations instantly. The first 60 minutes of lessons with Lessi are free with an account, including interactive practice.

Still have a question? Ask Lessi

Any question about this subject. Lessi answers right here, step by step.

Try asking

Asking needs a free account; your question is kept while you sign up. Answers are written by AI and can contain mistakes. Lessi doesn’t give medical or legal advice.

Learn it with a teacher

Learn this topic live with Lessi

Lessi, the AI teacher, explains it out loud on a board, with short quizzes, at your pace. The first 60 minutes are free.

Prefer a private teacher?

Private teachers for Mathematics

Teachers on Lessoons, online or face to face. Each profile opens on the Lessoons site.

See all 5,572 teachers