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GED math lesson

Point-Slope Form: Write the Equation, Then Turn It Into y = mx + b

The formula

y − y₁ = m(x − x₁)

m is the slope, and (x₁, y₁) is one point you know on the line. The x and y stay as letters, because they stand for every other point on the line.

(x₁, y₁)slope m(x, y)

You don't need to know where the line crosses the y-axis. One point and the slope are enough. Fill them in, multiply out the parentheses, get y by itself, and you have y = mx + b.

Practice 5 questions

Coach Crissy

Math instructor

Published
October 20, 2026
Reading time
6 min read

Why the point-slope formula works

Point-slope form is the slope formula with one point left as letters.

Start with the point you know, (x₁, y₁). Then picture any other point on the line and call it (x, y). From one to the other, the rise is y − y₁ and the run is x − x₁.

(x₁, y₁)(x, y)run = x − x₁rise = y − y₁
From the point you know to any other point on the line, the rise is y − y₁ and the run is x − x₁.

Slope is rise ÷ run, so m = (y − y₁) ÷ (x − x₁). Multiply both sides by (x − x₁) and you get y − y₁ = m(x − x₁). That’s the formula on your sheet.

Bottom line: one point and a slope are all you need to write the equation of a line.

Example 1: a taxi fare

You know what one ride cost and the price per mile, but not the flat fee. Here’s how it works:

Worked example
Point and slope

A taxi charges $2 a mile plus a flat fee. A 6-mile ride costs $17. Write an equation for the fare, y, after x miles. What is the flat fee?

The working

  1. The price per mile is the slope: m = 2
  2. The ride you know is the point (miles, dollars): (6, 17)
  3. Put them into the formula: y − 17 = 2(x − 6)
  4. Multiply out the parentheses: y − 17 = 2x − 12
  5. Add 17 to both sides: y = 2x + 5
  6. The 5 is the fare for 0 miles, so the flat fee is $5. Check: 2 × 6 + 5 = 17

Answer: y = 2x + 5, so the flat fee is $5

Steps 4 and 5 are how you get from point-slope form to y = mx + b every time. Multiply the slope into the parentheses. Then add or subtract to get y by itself.

Example 2: a line through two points

No slope given? Find it first with the slope formula. Then it’s the same steps as Example 1.

Worked example
Two points

Write the equation of the line through (−2, 7) and (3, −3) in the form y = mx + b.

The working

  1. Find the slope: m = (−3 − 7) ÷ (3 − (−2)) = −10 ÷ 5 = −2
  2. Put the slope and (−2, 7) into the formula: y − 7 = −2(x − (−2))
  3. Two minuses make a plus: y − 7 = −2(x + 2)
  4. Multiply out the parentheses: y − 7 = −2x − 4
  5. Add 7 to both sides: y = −2x + 3
  6. Check with the other point: −2(3) + 3 = −3. That matches (3, −3).

Answer: y = −2x + 3

The working in the practice set takes a shortcut. It puts the slope and one point straight into y = mx + b and solves for b. It’s the same math in a different order, and it gives the same equation.

Example 3: work backward from the equation

Some questions give you the equation and ask for the slope or a point. Line it up with the formula.

Worked example
Work backward

A line has the equation y + 4 = 3(x − 1). What is its slope, and what point on the line does the equation show?

The working

  1. Line it up with the formula: y − y₁ = m(x − x₁)
  2. The number in front of the parentheses is the slope: m = 3
  3. x − 1 matches x − x₁, so x₁ = 1
  4. y + 4 is the same as y − (−4), so y₁ = −4
  5. Check: put in x = 1. Then y + 4 = 3 × 0 = 0, so y = −4.

Answer: A slope of 3, through the point (1, −4)

Notice that the signs flip. A plus in the equation means the coordinate is negative.

3 mistakes that cost points

Mistake 1: keeping the wrong sign

With the point (4, −3), the formula starts y − (−3). Two minuses make a plus, so it’s y + 3 = m(x − 4). Writing y − 3 puts the whole line 6 units too high.

Mistake 2: multiplying only the x

The slope multiplies both numbers in the parentheses. In the taxi example, 2(x − 6) is 2x − 12. Stop at 2x − 6 and you get y = 2x + 11, an $11 flat fee when the real one is $5.

Mistake 3: swapping x₁ and y₁

In the point (4, 1), x₁ is 4 and y₁ is 1. With a slope of 2, that’s y − 1 = 2(x − 4), or y = 2x − 7. Swap them and you get y − 4 = 2(x − 1), which is y = 2x + 2, a different line.

Practice: 5 new questions

Every set is new. Check each answer as you go, and read the working on any you miss before the next one.

Writing the equation of a line through two points: 5 questions

Questions
5
Timer
None
No calculator
None

New questions every time you start. Do this set as many times as you like.

  • Check each answer as you go and read the working before the next question.
  • The formula sheet and calculator sit in the toolbar, as they do on the real test.

No account or email needed.

Want more? Practice all of functions and graphing (25 questions, every skill in the topic), or take the full GED math practice test.

Frequently asked questions

How do you turn 2 points into point-slope form?

Find the slope first, then use either point. For (1, 2) and (3, 8), m = (8 − 2) ÷ (3 − 1) = 3. Using (1, 2) gives y − 2 = 3(x − 1). Using (3, 8) gives y − 8 = 3(x − 3). They look different, but both are right. Each one becomes y = 3x − 1.

How can point-slope form be used in real life?

Use it when you know one moment and a steady rate, but not the starting amount. Say you've saved $70 after 4 weeks and you add $15 a week. That's y − 70 = 15(x − 4), or y = 15x + 10. So you started with $10, and by week 10 you'll have $160.

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Written by

Coach Crissy

Math instructor

Coach Crissy teaches the video lessons behind the BooknBoost GED math course, working through GED-style problems on camera: ratios and proportions, percent and interest, mean, median and mode, slope, and the geometry students meet on test day.