System of equations calculator

A system of equations calculator finds the values that make two or more equations true at the same time. Type two equations in x and y, or three in x, y and z, choose elimination or substitution, and our solver gives the exact answer with every step and a graph of where the lines meet. It also solves a line with a curve, such as y = x² or a circle.

In the UK these are called simultaneous equations, which is the name of our solver. It opens on 3x + 2y = 16 and 5x − 3y = 14, solved by elimination to give x = 4 and y = 2.

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Simultaneous Equations Solver

Solve linear and quadratic systems by elimination or substitution

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How to use the system of equations calculator

1. Choose 2 unknowns for x and y, or 3 unknowns for x, y and z.
2. Type each equation in its box, such as 2x+3y=12. For a power, type ^ and the number, then press the right arrow key to come back down.
3. Choose Elimination or Substitution for the working.
4. Read the answer and the steps. Each step says what it does, such as "Multiply (2) by 3", and the last one checks the answer in the first equation.
5. Look at the graph. For 2 unknowns it shows both lines or curves and where they cross. For 3 unknowns it shows the three planes.

Answers are exact, so a solution such as x = 7/3 stays a fraction, with a decimal alongside. If the lines are parallel or the same line, the calculator says so instead of giving numbers. The Examples list loads one of each kind, from parallel lines to a line and a circle. For practice, Practice and help sets random systems with worked solutions.

Solving a system by elimination

Elimination adds or subtracts the equations to remove one letter. Make the number in front of that letter the same size in both equations first.

Example 1: multiply one equation, then add

Solve 2x + 3y = 12 and 4x − y = 10.

1. Multiply the second equation by 3 so the y terms match in size: 12x − 3y = 30.
2. The signs of the y terms differ, so adding removes y: (2x + 3y) + (12x − 3y) = 12 + 30, which gives 14x = 42.
3. Divide by 14: x = 3.
4. Put x = 3 into the first equation: 6 + 3y = 12, so y = 2.

The solution is x = 3, y = 2, which you can write as the point (3, 2). On a graph, that is where the two lines cross:

Type both equations into the calculator with Elimination chosen, and the steps match these. Its graph draws the same two lines.

Solving a system by substitution

Substitution rearranges one equation to make a letter the subject, then puts that expression into the other equation. It is quickest when one equation already reads y = something.

Example 2: substitution

Solve y = 3x − 4 and 2x + y = 11. Replace y in the second equation with 3x − 4:

Then y = 3(3) − 4 = 5. The solution is x = 3, y = 5. Choose Substitution in the calculator to see these steps. Now try one in the course, then press Mark to check it:

A system of equations from a word problem

Example 3: drinks at a café

A café sells 15 drinks: coffees at $4 each and teas at $2 each. It takes $46. How many of each did it sell? Let x be the number of coffees and y the number of teas:

Multiply the first equation by 2 to get 2x + 2y = 30, then subtract it from the second: 2x = 16, so x = 8 and y = 7. The café sold 8 coffees and 7 teas. Check: 8 × 4 + 7 × 2 = 46. Typing x+y=15 and 4x+2y=46 into the calculator gives the same.

3 variable system of equations calculator

Choose 3 unknowns to solve three equations in x, y and z. Elimination removes one letter twice to leave two equations in two letters, solves those, then works back.

Example 4: three unknowns

Solve x + y + z = 9, 2x − y + 3z = 13 and x + 2y − z = 4.

1. Remove z from the first two: 3 × (1) − (2) gives x + 4y = 14.
2. Remove z from the first and third: (1) + (3) gives 2x + 3y = 13.
3. Solve those two: 2 × (x + 4y = 14) − (2x + 3y = 13) gives 5y = 15, so y = 3, and then x = 2.
4. Put x = 2 and y = 3 into the first equation: z = 9 − 2 − 3 = 4.

The solution is x = 2, y = 3, z = 4. The calculator follows the same order and draws the three planes meeting at that point. Three equations can also be solved with matrices; the Matrix Calculator finds inverses and determinants for that method.

No solution or infinitely many solutions

Not every system has exactly one answer. The calculator spots the other two cases and says which you have.

A quick check for two linear equations: if the slopes are the same, the lines are parallel or identical. Rearranged into y = mx + b form, the first pair both have slope 2 but different intercepts, so there is no solution.

A line and a curve

The calculator also solves a linear equation with a quadratic one, such as a line and a parabola or a line and a circle. Substitution turns the pair into a single quadratic, which can have two, one or no solutions.

Example 5: a line and a parabola

Solve y = 4x − 3 and y = x². Set them equal and factor:

So x = 1 or x = 3, giving the points (1, 1) and (3, 9). The calculator says the line crosses the curve at two points. If the line just touches the curve, it calls it a tangent; if they never meet, there is no real solution. Our simultaneous equations guide goes further into each method.

Common mistakes

• Multiplying only one side. When you multiply an equation to match a letter, multiply every term, including the number on the right.
• Adding when you should subtract. Subtract when the matched letters have the same sign, add when the signs are opposite.
• Stopping after one letter. A system needs every unknown. Put your first answer back in to find the other.
• Losing a negative in substitution. Put the expression in brackets: 2x + (3x − 4), not 2x + 3x − 4 with signs dropped later.
• Not checking. Put both values into the equation you did not use last. It takes seconds and catches most slips.
• Calling 0 = 0 "no solution". 0 = 0 means infinitely many solutions. A false statement such as 0 = −4 means none.

Practise systems of equations in a short timed paper

Systems of equations come up in Pre-Algebra & Algebra 1 and Geometry, Algebra 2 & SAT Math practice, and as simultaneous equations in GCSE Maths Higher in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. The Algebra course shows where two lines meet before the practice questions.

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Maths · AlgebraSimultaneous equations: how to solve them, with examplesHow to solve simultaneous equations by elimination, substitution and graphs, with worked GCSE examples, word problems and practice questions with answers.