Linear Equation Solver

Solve systems of linear equations

A linear equation is an algebraic equation of the form ax + b = c (or a system of such equations) whose graph is a straight line, solved by isolating the variable using arithmetic operations.

System of 2 Equations (2 Variables)

x + y =

Solution

Single Linear Equation

Solve ax + b = c for x

x + =

Solution

Line Equation (Two Points)

Point 1: ( , )

Point 2: ( , )

Line Equation: y = mx + b

slope (m)

y-intercept (b)

Frequently Asked Questions

How do you solve a system of two linear equations with two variables?

The calculator uses Cramer’s Rule (determinant method). For 2x + 3y = 8 and 4x − y = 2, compute the determinant D = (2)(−1) − (4)(3) = −14. Then x = (8×(−1) − 2×3)/(−14) = 1 and y = (2×2 − 4×8)/(−14) = 2. Verify: 2(1) + 3(2) = 8 ✓.

What does it mean when the linear system solver returns N/A?

N/A means the determinant is zero, indicating the two equations are either parallel (no solution) or identical (infinitely many solutions). For example, x + y = 3 and 2x + 2y = 6 are the same line — every point on it satisfies both equations, so there is no unique solution.

How does the calculator find the slope between two points?

Slope m = (y₂ − y₁)/(x₂ − x₁). For points (1, 2) and (3, 6): m = (6 − 2)/(3 − 1) = 2. Then the y-intercept b = y₁ − m × x₁ = 2 − 2(1) = 0, giving the line y = 2x. If x₁ = x₂, the line is vertical (undefined slope) and the calculator displays ∞.

What is the difference between a single linear equation and a system of linear equations?

A single linear equation (ax + b = c) has one variable and always yields one solution: x = (c − b)/a, provided a ≠ 0. A system has two or more equations with two or more unknowns; it requires finding values that satisfy all equations simultaneously, using methods like substitution, elimination, or matrix algebra.

What happens if a = 0 in a single linear equation ax + b = c?

If a = 0, there is no x term. The equation becomes b = c. If b = c, every value of x is a solution (infinite solutions). If b ≠ c, there is no solution. The calculator returns N/A for a = 0 because a unique solution cannot be determined.