Curriculum fit: US Grade 8 and Algebra 1 (Common Core 8.EE.B and Functions). The same idea is called y = mx + c in the UK; see the straight line graphs guide for the GCSE version.
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What Is It?
Slope-intercept form writes the equation of a straight line as y = mx + b. The number m is the slope, which tells you how steep the line is and which way it leans. The number b is the y-intercept, the point where the line crosses the y-axis.
The slope is the rise divided by the run. The rise is how far the line goes up or down and the run is how far it goes across. A positive slope goes up from left to right, a negative slope goes down, and a slope of zero is flat. Many real situations fit this form: b is the starting amount and m is the rate of change.
Key Formulas and Ideas
- Slope-intercept form: y = mx + b, where m = slope and b = y-intercept (0, b).
- Slope from two points: m = (y₂ − y₁) ÷ (x₂ − x₁).
- Standard form Ax + By = C becomes y = −(A/B)x + C/B (when B ≠ 0).
- Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals, so their product is −1.
- A horizontal line y = b has slope 0. A vertical line x = a has undefined slope.
Worked Examples
Graph a Line From Its Equation
Graph y = −2x + 3.
- The y-intercept is b = 3, so plot the point (0, 3).
- The slope is −2, which is −2/1: go down 2 and right 1.
- From (0, 3) that gives (1, 1), and again gives (2, −1).
- Draw a straight line through the points.
✓ Answer A line through (0, 3), (1, 1) and (2, −1)
Write an Equation From Two Points
Write the equation of the line through (2, 5) and (4, 11).
- Find the slope: m = (11 − 5) ÷ (4 − 2) = 6 ÷ 2 = 3.
- Put one point into y = 3x + b: 5 = 3(2) + b, so b = −1.
- Write the equation: y = 3x − 1.
- Check with (4, 11): 3(4) − 1 = 11. ✓
✓ Answer y = 3x − 1
Convert From Standard Form
Write 3x + 4y = 12 in slope-intercept form and state the slope and y-intercept.
- Subtract 3x from both sides: 4y = −3x + 12.
- Divide every term by 4: y = −(3/4)x + 3.
- Compare with y = mx + b.
✓ Answer y = −(3/4)x + 3, so the slope is −3/4 and the y-intercept is 3
Common Mistakes to Avoid
- Writing the slope as run ÷ rise instead of rise ÷ run.
- Losing the sign on a negative slope, or subtracting the coordinates in a different order on the top and bottom.
- Reading b straight from standard form. In 3x + 4y = 12 the y-intercept is 3, not 12.
- Dividing only some terms by B when rearranging. Every term must be divided.
- Mixing up zero slope (a flat line) and undefined slope (a vertical line).
Slope-Intercept Form in Word Problems
In a word problem, b is the starting value and m is the rate of change. A gym charges a $20 joining fee and $12 per month. The cost after x months is y = 12x + 20. The y-intercept 20 is the joining fee and the slope 12 is the cost per month. After 6 months the cost is 12(6) + 20 = $92.
Parallel and Perpendicular Lines
To write a line parallel to y = 2x + 1 through (0, 4), keep the slope 2 and use b = 4: y = 2x + 4. For a perpendicular line, flip the slope and change its sign to get −1/2, so y = −(1/2)x + 4.
Practice Questions
Frequently Asked Questions
What is slope-intercept form?
It is the equation of a line written as y = mx + b, where m is the slope and b is the y-intercept. It shows you straight away where to start on the graph and how to move to the next point.
What is the difference between slope-intercept form and point-slope form?
Slope-intercept form is y = mx + b and needs the y-intercept. Point-slope form is y − y₁ = m(x − x₁) and starts from any point on the line. Both describe the same line and you can convert between them.
Is y = mx + b the same as y = mx + c?
Yes. US textbooks write y = mx + b and UK textbooks write y = mx + c. In both, m is the gradient or slope and the last term is the y-intercept.
Keep Going With These Guides
Straight Line Graphs: Gradient and y = mx + c
Plot straight lines, work out the gradient and read the equation from a graph. Covers y = mx + c (slope-intercept form), with worked examples.
GCSEHigh SchoolFind the Equation of a Line From Two Points
Work out the gradient (slope), find the y-intercept and write the equation of a line, plus parallel and perpendicular lines and midpoints.
GCSEA-LevelHigh SchoolSolving Linear Equations Step by Step (With Examples)
Learn the balance method for solving linear equations, including brackets and x on both sides, with worked examples for GCSE and Algebra 1.
KS3GCSEMiddle SchoolHigh SchoolSimultaneous Equations (Systems of Equations) Step by Step
Solve two equations with two unknowns using elimination and substitution. Worked examples for GCSE simultaneous equations and Algebra 1 systems.
GCSEHigh SchoolSources and Review
Reviewed by StudyMorrow editorial review on 7 Nov 2026.