Linear Functions
The Four Forms of a Straight Line
Slope-intercept, point-slope, standard and intercept form — what each is for, and how to move between them.
What this page covers
- Write a line in each of the four standard forms
- Choose the form that suits the information given
- Convert between forms
- Read the slope and intercepts from an equation
The four forms
| Form | Equation | Given | Shows directly |
|---|---|---|---|
| Slope-intercept | y = mx + b | Slope and y-intercept | Slope m, intercept (0, b) |
| Point-slope | y - y0 = m(x - x0) | Slope and any point | Slope and the point used |
| Standard | Ax + By = C | Anything | Handles vertical lines; used in systems |
| Intercept | xa + yb = 1 | Both intercepts | Intercepts (a, 0) and (0, b) |
They describe the same lines. The choice is about which information is to hand and which is to be read off.
Slope-intercept form
The source derives the meaning of b rather than asserting it: set x = 0 and the equation collapses.
It also identifies the slope by inspection. For y = 2x - 3 the source notes the number 2 is the slope, and the intercept is at y = -3.
| Equation | Slope | y-intercept |
|---|---|---|
| y = 2x - 3 | 2 | (0, -3) |
| y = 5 - 4x | -4 | (0, 5) |
| y = 12x | 12 | (0, 0) |
| y = 7 | 0 | (0, 7) |
The source flags that y = 5 - 4x is the same as y = -4x + 5: the slope is -4, not 5. Rewriting into mx + b order before reading off avoids the confusion.
If the y-intercept is 0 the line passes through the origin, which the source records for the case b = 0.
Point-slope form
It is the slope formula rearranged. If (x, y) is any other point on the line, then
This is the form to reach for first. Two points give a slope and either point serves as (x0, y0), so it is always usable.
Worked example — the source's case
Line through (-2, 3) with slope -12.
At x = -2: y = 1 + 2 = 3 ✓.
From two points — the source's case
Line through (4, -2) and (2, 5).
7(4) + 2(-2) = 24 ✓; 7(2) + 2(5) = 24 ✓. Checking both given points is the right test, since either one alone would pass for a wrong slope.
Standard form
Conventionally the coefficients are integers with no common factor, and A is taken positive. The source demonstrates the conversion:
Handles vertical lines
x = 3 is 1 · x + 0 · y = 3. Slope-intercept form cannot express it at all.
Symmetric in x and y
Neither variable is privileged, which suits systems of equations.
Matches matrix methods
A system in standard form maps directly onto an augmented matrix.
Intercept form
Worked example — the source's case
Line through (3, 0) and (0, 4).
Clearing the denominators by multiplying through by 12 gives standard form:
At (3, 0): 12 = 12 ✓. At (0, 4): 12 = 12 ✓.
Intercept form requires a right-hand side of exactly 1 and neither intercept to be zero. A line through the origin cannot be written this way, since a or b would be zero.
Converting between forms
| From | To | Method |
|---|---|---|
| Two points | Point-slope | Compute m, use either point |
| Point-slope | Slope-intercept | Expand and solve for y |
| Slope-intercept | Standard | Move the x term across, clear fractions, make A positive |
| Standard | Slope-intercept | Solve for y; the slope is -AB |
| Standard | Intercept | Divide through by C, provided C ≠ 0 |
| Intercept | Standard | Multiply by the common denominator |
From Ax + By = C the slope is -AB and the y-intercept is CB, without rearranging. For 7x + 2y = 24 the slope is -72, matching the worked example above.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Reading the slope of y = 5 - 4x as 5 | It is -4 | Rewrite as -4x + 5 first |
| y - 3 = m(x + 2) written as y + 3 = m(x - 2) | Both signs are wrong | Substitute the point back in |
| Leaving fractions in standard form | Clear them | Multiply through by the common denominator |
| Using slope-intercept for a vertical line | Use x = c | A vertical line has no slope |
| Checking only one of two given points | Check both | One point passes for many wrong lines |
| Intercept form with right side not 1 | Divide through | xa + yb = 1 exactly |
Frequently asked questions
Which form should I start from?
Point-slope, almost always. Any two points give a slope and a point, which is exactly what it needs. Convert afterwards to whatever the question wants.
Why does standard form exist?
It handles vertical lines, which slope-intercept cannot, and it is the form systems of equations are written in. x = 3 is a perfectly good line with no slope-intercept representation.
What does b mean in y = mx + b?
The y-intercept. Setting x = 0 gives y = b, so the line crosses the y-axis at (0, b). The source derives it exactly this way.
When is intercept form useful?
When both intercepts are known or wanted. x3 + y4 = 1 shows immediately that the line crosses at (3, 0) and (0, 4).
Source. Handwritten teaching notes, Week 5, pages 6-9.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.