Slope-intercept, point-slope, standard form
High School Essentials • 9-12
Linear equations are the foundation of algebra and appear everywhere -- from calculating costs and predicting trends to engineering and economics. A linear equation describes a straight line, and mastering the three standard ways to write one gives you the flexibility to tackle any problem involving linear relationships.
This is the most commonly used form because it immediately tells you two critical pieces of information:
| Component | Meaning | Example in y = 3x - 2 |
|---|---|---|
| m | Slope (steepness and direction) | m = 3 (rises 3 for every 1 right) |
| b | y-intercept (where the line crosses the y-axis) | b = -2 (crosses at (0, -2)) |
Given a slope and a y-intercept, you can write the equation instantly. For example, a line with slope -4 that crosses the y-axis at 7 is simply y = -4x + 7.
When you know the slope m and any single point (x1, y1) on the line, point-slope form is the fastest route to an equation. You plug the point and slope directly in -- no need to solve for b first.
Write the equation of the line through (2, 5) with slope m = -3.
Standard form uses integer coefficients with A ≥ 0 (and A, B not both zero). It is especially useful for finding intercepts quickly and for solving systems of equations.
To find intercepts from standard form:
Convert y = (2/3)x - 4 to standard form.
The standard form is 2x - 3y = 12.
Convert 5x + 2y = 10 to slope-intercept form.
The slope is -5/2 and the y-intercept is 5.
When moving terms across the equals sign, students often forget to change the sign. If you have y - 5 = -3(x - 2), distributing the -3 gives -3x + 6, not -3x - 6. Always distribute carefully before simplifying.
Use slope-intercept when you need to graph quickly or know the slope and y-intercept. Use point-slope when given a point and a slope (or two points). Use standard form when working with systems of equations or when the problem asks for integer coefficients.
1. Write the equation of the line with slope 4 and y-intercept -1 in slope-intercept form.
y = 4x - 1. Plug m = 4 and b = -1 directly into y = mx + b.
2. Write the equation of the line through (-1, 3) with slope 2 in point-slope form, then convert to slope-intercept form.
Point-slope: y - 3 = 2(x - (-1)), which simplifies to y - 3 = 2(x + 1).
Slope-intercept: y - 3 = 2x + 2, so y = 2x + 5.
3. Convert y = -3x + 9 to standard form.
Move -3x to the left: 3x + y = 9. This is already in standard form with A = 3, B = 1, C = 9.
4. Convert 4x - 6y = 24 to slope-intercept form.
-6y = -4x + 24. Divide by -6: y = (2/3)x - 4. The slope is 2/3 and the y-intercept is -4.
5. A line passes through (3, -2) and (6, 7). Find the equation in slope-intercept form.
First find the slope: m = (7 - (-2)) / (6 - 3) = 9/3 = 3.
Use point-slope with (3, -2): y - (-2) = 3(x - 3), so y + 2 = 3x - 9, giving y = 3x - 11.
Linear equations can be expressed in three forms: slope-intercept (y = mx + b), point-slope (y - y1 = m(x - x1)), and standard form (Ax + By = C). Each form has strategic advantages. Converting between them requires careful algebraic manipulation -- distribute, combine like terms, and keep track of signs. Mastering all three forms prepares you for graphing, systems, and real-world modeling.