Calculators / Algebra 1 / System of Equations Solver Calculator

System of Equations Solver Calculator

Published on: December 1, 2024

This System of Equations Solver Calculator helps you solve two equations with two variables step by step. The goal is to find the values that make both equations true at the same time. You can solve the system by methods such as substitution or elimination, then check the solution in both equations. It is a simple way to check answers, understand system solving, and practise basic algebra step by step.

Step-by-step method

  1. Identify the two equations in the system.
  2. Rewrite both equations in standard form.
  3. Multiply one equation to match a coefficient.
  4. Subtract the equations to eliminate one variable.
  5. Solve for the remaining variable.
  6. Substitute back to solve for the other variable.

Example 1: x + y = 3; 2x - y = 4

Step 1 - Identify the two equations in the system.

In this problem: These are the two equations in the system.

Eq 1: x + y = 3Eq 2: 2x - y = 4

Step 2 - Rewrite both equations in standard form.

In this problem: Move everything to the left side so each equation equals 0.

Eq 1: x + y - 3 = 0Eq 2: 2x - y - 4 = 0

Step 3 - Multiply one equation to match a coefficient.

In this problem: Multiply Equation 1 so the coefficient of x matches Equation 2.

(x + y - 3) × (2) = 2x + 2y - 6

Step 4 - Subtract the equations to eliminate one variable.

In this problem: Subtract the equations to eliminate x.

(2x + 2y - 6) − (2x - y - 4) = 3y - 2 = 0

Step 5 - Solve for the remaining variable.

In this problem: Solve for y.

y = 2/3

Step 6 - Substitute back to solve for the other variable.

In this problem: Substitute y = 2/3 into an original equation to solve for x.

x + 2/3 = 3 ⇒ x = 7/3

Final answer: x = 7/3, y = 2/3

Example 2: 3x + 2y = 12; x - y = 1

Step 1 - Identify the two equations in the system.

In this problem: These are the two equations in the system.

Eq 1: 3x + 2y = 12Eq 2: x - y = 1

Step 2 - Rewrite both equations in standard form.

In this problem: Move everything to the left side so each equation equals 0.

Eq 1: 3x + 2y - 12 = 0Eq 2: x - y - 1 = 0

Step 3 - Multiply one equation to match a coefficient.

In this problem: Multiply Equation 1 so the coefficient of x matches Equation 2.

(3x + 2y - 12) × (1/3) = x + 2y/3 - 4

Step 4 - Subtract the equations to eliminate one variable.

In this problem: Subtract the equations to eliminate x.

(x + 2y/3 - 4) − (x - y - 1) = 5y/3 - 3 = 0

Step 5 - Solve for the remaining variable.

In this problem: Solve for y.

y = 9/5

Step 6 - Substitute back to solve for the other variable.

In this problem: Substitute y = 9/5 into an original equation to solve for x.

3x + 18/5 = 12 ⇒ x = 14/5

Final answer: x = 14/5, y = 9/5