System of Equations Solver

How it works

Each equation goes in as ax + by = c. The calculator multiplies the equations so the x terms match, subtracts to get y, then puts y back in to get x. Parallel lines give no solution, and two copies of one line give infinitely many.

Example

Solve 2x + 3y = 12 and x − y = 1.

x = 3, y = 2

Steps

1. 1
   The equations: 2x + 3y = 12 and x − y = 1.
2. 2
   Multiply the first equation by 1 and the second by 2, so both x terms match: 2x + 3y = 12 and 2x − 2y = 2.
3. 3
   Subtract to remove x: −5y = −10, so y = 2.
4. 4
   Put y into the first equation: 2x + 6 = 12, so x = 3.
5. 5
   Check the second equation: 1 = 1.

More examples

Solve 3x − 2y = 4 and 5x + 4y = 14.

x = 2, y = 1

Steps

1. 1
   The equations: 3x − 2y = 4 and 5x + 4y = 14.
2. 2
   Multiply the first equation by 5 and the second by 3, so both x terms match: 15x − 10y = 20 and 15x + 12y = 42.
3. 3
   Subtract to remove x: 22y = 22, so y = 1.
4. 4
   Put y into the first equation: 3x − 2 = 4, so x = 2.
5. 5
   Check the second equation: 14 = 14.

Solve x + y = 4 and 2x + 2y = 10.

No solution

Steps

1. 1
   The equations: x + y = 4 and 2x + 2y = 10.
2. 2
   The x and y coefficients are in the same ratio, so the lines are parallel.
3. 3
   The constants don’t match that ratio: the lines never meet.

Common mistake

Line up the terms first. For y = 2x + 1, rewrite it as −2x + y = 1, so a = −2, b = 1 and c = 1.

Questions

Elimination or substitution?

Substitution is quicker when a variable already stands alone, like y = 2x + 1. Elimination is quicker when the coefficients match or are easy to match. Both give the same answer.

How do I check the answer?

Put x and y back into both original equations. Both have to come out true.

What does it look like on a graph?

Each equation is a line, and the solution is where they cross. Parallel lines never cross, so there is no solution.

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