Linear Equations

9 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team

A linear equation is one where the variable appears only to the first power. Its graph is a straight line, and solving it takes a few reliable steps. Linear equations are everywhere: pricing, speed, conversions and rates of change all follow them.

Solving a linear equation in one variable

Isolate x by undoing operations. For 3x − 5 = 4x + 2: subtract 3x from both sides to get −5 = x + 2, then subtract 2 to get x = −7. Check: 3(−7) − 5 = −26 and 4(−7) + 2 = −26 ✓. When fractions appear, multiply every term by the common denominator first.

Slope-intercept form: y = mx + b

The slope m tells you how steep the line is (rise over run) and b is where it crosses the y-axis. y = 2x − 1 rises 2 for every 1 to the right and crosses the y-axis at −1. See it in the graphing calculator.

Finding the equation from two points

Through (2, 3) and (6, 11): slope = (11 − 3) ÷ (6 − 2) = 2. Substitute a point into y = 2x + b: 3 = 4 + b, so b = −1. The line is y = 2x − 1. Check with the second point: 2(6) − 1 = 11 ✓.

Parallel and perpendicular lines

Parallel lines have equal slopes. Perpendicular lines have slopes that multiply to −1 (negative reciprocals): the line perpendicular to y = 2x − 1 has slope −1/2.

Inequalities

Solve like an equation, but flip the inequality sign when you multiply or divide by a negative number. 3x − 5 > 10 gives x > 5. And −2x < 6 gives x > −3.

Systems of two equations

Substitution: y = 2x + 1 and 3x + y = 11. Replace y: 3x + 2x + 1 = 11, so x = 2 and y = 5. Elimination: add or subtract the equations to remove a variable. The equation solver does both types and flags no-solution cases.

Practice questions

Test yourself, then tap to check. Answers are calculated by the tools linked below.

Solve 4x − 3 = x + 12.
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x: 5
Find the roots of f(x) = x² − 6x + 8.
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Function: f(x) = x^2 - 6x + 8
Solve x² − 7x + 10 = 0.
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x₁: 5 · x₂: 2