Simultaneous Equations Calculator
Free math practice app for middle school students
Enter two linear equations to calculate the values of x and y.
Add or subtract the two equations to eliminate one variable and solve for the other.
(Example)① x+y=5 ② 2x+y=7
How to Use the Calculator
Enter two equations
Type the two simultaneous equations into fields ① and ②. Example: x+y=5 and 2x+y=7
Choose a solution method
Select either Elimination or Substitution. Both methods produce the same answer.
Calculate and review the result
Click Calculate to display x and y. Use Clear if you want to enter new equations.
How to Solve Simultaneous Equations
Finding the values of x and y from two equations is called solving simultaneous equations.
No matter which method you choose, the goal is the same: eliminate one variable and solve for the other.
Elimination and substitution lead to the same answer. Choose the method that feels easier for the problem.
Elimination Method
Match the coefficients, then add or subtract the equations to eliminate one variable.
① 5x + 3y = 11
② 2x + y = 5
②×3 → 6x + 3y = 15
(②×3) - ① → x = 4
Substitute into ② → 2×4 + y = 5 → y = -3
Once the coefficients match, you can remove one variable using only addition or subtraction.
Substitution Method
Rewrite one equation as x=... or y=..., then substitute that expression into the other equation.
① 2x + y = 6
② 2x + 5y = 10
From ①, 2x = 6 - y
Substitute into ② → (6-y)+5y=10
4y = 4 → y = 1
Substitute into ① → 2x + 1 = 6 → x = 2.5
By rewriting one equation first, you can turn the system into a one-variable equation.
A=B=C Type Equations
If an expression is chained like A=B=C, split it into two equations first, then solve the system.
2x + 7y = 3x + 5y = 22
→ 2x + 7y = 22
→ 3x + 5y = 22
Solve these two equations using elimination or substitution
Example Problems
0.4x + (5/6)x - y = 2/3, 0.5x - 0.15y = 0.25
A:
Simplify to 30x - 12y = 10
Simplify to 50x - 15y = 25
Match coefficients and subtract → y = 5/3
Substitute back to get x = 1
(2/3)x + y = x + 2, 2(3x - 2 - y) - (x + 2y) = 3
A:
Simplify the first equation to 2x - 3y = 6
Simplify the second equation to 5x - 6y = 3
Substitute to get y = 11/3
Substitute back to get x = 5
(2/3)(4x - 3y) = 20, 4x - 5y = 20
A:
Simplify the first equation to 8x - 6y = 60
Multiply the second equation by 2 → 8x - 10y = 40
Subtract to get y = 5
Substitute back to get x = 45/4
Where You Might Use This
- 1Find unit prices from the total cost and quantity of purchases
- 2Work out distance from relationships between speed and time
- 3Solve mixing problems by finding two unknown quantities
- 4Reverse-calculate test scores from correct and incorrect answers
Examples of Invalid Input
The following inputs cannot be calculated. Please enter the equations in a valid format.
Cases with empty input
①: (blank) / ②: x+y=5
The first equation has not been entered.
Enter an equation in both fields.
①: 2x+y=7 / ②: (blank)
The second equation has not been entered.
Enter both equations before calculating.
①: (blank) / ②: (blank)
Neither equation has been entered.
Enter the two simultaneous equations.
Cases with an incorrect equals sign
①: x+y / ②: 2x+y=7
The first equation does not contain '='.
Make sure each equation contains one '=' sign, for example x+y=5.
①: 2x+y=5=3 / ②: x-y=1
The first equation contains more than one '=' sign.
Each equation should contain only one '=' sign.
①: x+2y / ②: 3x-y
Neither equation contains '='.
Add one '=' sign to each equation.
Cases with an invalid equation format
①: 5=5 / ②: x+y=3
The first equation does not contain a variable.
Enter a linear equation that includes x or y.
①: abc+y=5 / ②: x+y=3
An unsupported variable name is included. Only x and y are allowed.
Use only x and y as variables.
①: x++y=5 / ②: x+y=3
The operator syntax is invalid.
Use valid operators such as + and -.
Examples that are not errors
The following inputs are valid. Even when there is no solution or infinitely many solutions, the calculator shows that result instead of an error.
x+y=5, 2x+y=7 → x=2, y=3
x+y=5, x+y=10 → no solution
x+y=5, 2x+2y=10 → infinitely many solutions
x=3, y=4 → x=3, y=4
Precautions
- Enter linear equations using x and y, for example 2x+3y=7
- Quadratic expressions such as x² and equations with more than two variables are not supported
- Use only one '=' sign per equation
- If there is no solution or infinitely many solutions, the calculator will tell you
- Decimal input is allowed, but rounding may sometimes produce decimal output
Frequently Asked Questions
Q1: Do elimination and substitution give different answers?
A: No. If the calculations are done correctly, both methods give the same solution.
Q2: Can I use equations with decimals or fractions?
A: Yes. Decimals are supported, and you can enter fractions after converting them to decimals.
Q3: Why does the calculator sometimes show no solution?
A: That happens when the two equations represent parallel lines, so they never intersect.
Q4: What happens if I enter the same equation twice?
A: If the two equations are equivalent, the calculator will show infinitely many solutions.
Q5: Does this support three-variable systems?
A: Not at the moment. It currently supports only two-variable linear systems using x and y.