Algebra

Simultaneous Equations Solver

Solve two simultaneous equations with two unknowns using the elimination method. Enter the coefficients of both equations and the calculator finds x and y with full step-by-step working — matching coefficients, eliminating a variable, and substituting back.

Equation 1: 2x + 3y = 13

a₁ (x coeff)

b₁ (y coeff)

c₁ (right side)

Equation 2: 3x -1y = 4

a₂ (x coeff)

b₂ (y coeff)

c₂ (right side)

Solution

2.272727

2.818182

How it works

Write the simultaneous equations
$(1)\; 2x+3y = 13 \qquad (2)\; 3x-y = 4$
We have two equations with two unknowns, x and y. We will use the elimination method.
Multiply equations to match x coefficients
$(1) \times 3: \; 6x+9y = 39 \qquad (2) \times 2: \; 6x-2y = 8$
Multiply equation (1) by 3 and equation (2) by 2 so both have 6x.
Subtract equations to eliminate x
$+11y = 31$
Subtract the equations: the x terms cancel out.
Result:11y = 31
Divide both sides by 11
$y = \frac{31}{11} = 2.818182$
Divide both sides by 11 to find y.
Result:y = 2.818182
Substitute y into equation (1)
$2x +3(2.818182) = 13$
Substitute y = 2.818182 into equation (1): 2x + 8.454545 = 13.
Solve for x
$2x = 13 - 8.454545 = 4.545455$
2x = 4.545455, so x = 2.272727.
Result:x = 2.272727
Check in equation (2)
$3(2.272727) -1(2.818182) = 4$
Check: 3 × 2.272727 + -1 × 2.818182 = 4. Expected 4. ✓

The formula

a_1x + b_1y = c_1 \quad \text{and} \quad a_2x + b_2y = c_2

a₁, b₁, c₁

Coefficients of the first equation

a₂, b₂, c₂

Coefficients of the second equation

Example substitution

2x + 3y = 13 \quad 3x - y = 4 \implies x = 2.5, \; y = 2.67

Multiply equation 2 by 3 to match y coefficients, then add equations to eliminate y.

Worked examples

2x + y = 7 and x + y = 4

1
Subtract eq2 from eq1: x = 3
2
Substitute into eq2: 3 + y = 4
3
Solve for y: y = 1

Answer: x = 3, y = 1

3x + 2y = 12 and x − y = 1

1
From eq2: x = y + 1
2
Substitute: 3(y+1) + 2y = 12
3
Solve: 5y = 9, y = 1.8
4
x: x = 2.8

Answer: x = 2.8, y = 1.8

4x + 3y = 25 and 2x − y = 5

1
Multiply eq2 by 3: 6x − 3y = 15
2
Add equations: 10x = 40
3
x = 4:
4
y: y = 2(4) − 5 = 3

Answer: x = 4, y = 3

Frequently asked questions

How do you solve simultaneous equations?+

Use elimination (multiply equations to match one variable, then add or subtract) or substitution (rearrange one equation and substitute into the other).

What is the elimination method?+

Multiply one or both equations so that one variable has the same coefficient, then add or subtract the equations to eliminate that variable.

How do you check simultaneous equation answers?+

Substitute your values of x and y into both original equations. Both should balance.

What if the equations have no solution?+

If the equations represent parallel lines, there is no solution. If they represent the same line, there are infinitely many solutions.

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