Linear Equations Guide: Solve Them Step by Step

The direct answer: a linear equation is one whose graph is a straight line, and the skills break into four useful cases. One variable equations ask you to isolate x. Slope intercept form, y = mx + b, lets you read slope and the y intercept directly. Standard form, Ax + By = C, is handy for finding intercepts and for systems. A system of two equations is solved by substitution or elimination to find the point the lines share. This guide walks through each with worked examples.

Linear Equations at a Glance

One Variable Equations

The goal is to isolate x by doing the same operation to both sides. Khan Academy frames slope intercept form as giving the slope and the y intercept directly, which is the foundation for everything after (Khan Academy slope intercept intro).

Worked example. Solve 4x - 7 = 9.

Add 7 to both sides: 4x = 16. Divide by 4: x = 4. Check: 4 times 4 minus 7 = 16 - 7 = 9. Correct.

If a variable is on both sides, collect x terms on one side first. Solve 3x + 2 = x + 10. Subtract x: 2x + 2 = 10. Subtract 2: 2x = 8. Divide: x = 4.

Slope Intercept Form

The form y = mx + b describes a line. The m is the slope, the rate of change. The b is the y intercept, where the line crosses the y axis. Khan Academy notes that the form is called slope intercept precisely because it gives the slope and the intercept (Khan Academy slope intercept intro).

Worked example. For y = 2x + 1, the slope is 2 and the line crosses the y axis at (0, 1). For each step right, the line rises 2.

A slope of 0 is a flat horizontal line. An undefined slope is a vertical line. If you are reviewing the algebra basics behind these manipulations, our algebra basics guide covers the rules in more detail.

Graphing a Line

To graph y = 2x + 1, plot the y intercept (0, 1), then use the slope: up 2, right 1, to reach (1, 3). Draw the line through both points. Two points are enough to fix a straight line.

Standard Form and Intercepts

In Ax + By = C, find intercepts by setting the other variable to 0.

Worked example. 2x + 3y = 6.

Set x = 0: 3y = 6, so y = 2. Point (0, 2). Set y = 0: 2x = 6, so x = 3. Point (3, 0). Draw the line through (0, 2) and (3, 0).

Standard form is also the natural starting point for systems, because both equations share the same Ax + By = C shape.

Systems of Equations

A system has two equations. The solution is the point both lines share. Two common methods are substitution and elimination. Khan Academy's SAT lesson walks through both, noting the goal is always to reduce two equations with two variables down to one equation with one variable (Khan Academy solving systems).

Worked example by substitution. y = x + 1 and 2x + y = 7.

Substitute x + 1 for y in the second: 2x + (x + 1) = 7, so 3x + 1 = 7. Then 3x = 6, x = 2. From y = x + 1, y = 3. Solution (2, 3). Check in both equations: 3 = 2 + 1, and 4 + 3 = 7. Correct.

Worked example by elimination. x + y = 5 and x - y = 1.

Add the equations: 2x = 6, so x = 3. Then y = 2. Solution (3, 2).

Special cases matter. If elimination leads to a true statement like 0 = 0, the equations describe the same line (infinitely many solutions). If it leads to a false statement, the lines are parallel and there is no solution (Saylor solving linear systems). If you get stuck turning a word problem into one of these forms, our guide to word problems shows the setup steps.

Common Misconceptions

• "Add 7 to one side but not the other." Every operation must hit both sides.
• "Slope is b, not m." In y = mx + b, m is the slope and b is the intercept.
• "The y intercept is on the x axis." It is where the line crosses the y axis, at x = 0.
• "Check only one equation in a system." Check the solution in both.
• "Parallel lines eventually meet." Parallel lines have equal slope and never meet, so the system has no solution.

Frequently Asked Questions

What makes an equation linear?

Its graph is a straight line, and no variable is raised to a power higher than 1.

What is slope?

Slope is the rate of change, the rise over the run between two points on the line.

What is the y intercept?

It is the point where the line crosses the y axis, the value of y when x = 0.

How do I solve a system?

Use substitution or elimination to find the point both equations share.

What if two lines are parallel?

Parallel lines have equal slope and never meet, so the system has no solution.

When do students learn linear equations?

Usually in grade 8 or 9 as part of introductory algebra, with systems following soon after.