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Systems of Equations Solver

Enter each equation on a new line and follow the solution step by step.

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System of Equations Solver with Steps: Free Linear & Nonlinear Systems Calculator

A system asks for values that satisfy multiple equations at once. Math AI Solver helps you work through the algebra and check the shared result, whether you enter linear equations or a nonlinear example.

Find the shared values

Follow how two or more equations narrow down the values that satisfy them all.

Follow the algebra

See the chosen operations and check the answer against each original equation.

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Enter equations on separate lines or attach a clear photo of your question.

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The method

How to Solve Systems of Equations Step by Step

Start by arranging the equations, choose an algebraic path, and verify that every original equation holds.

1

Write the system

Put each equation on its own line. Rearrange linear equations into standard form when that makes the coefficients easier to compare.

a1x+b1y=c1a2x+b2y=c2a_1x+b_1y=c_1 \qquad a_2x+b_2y=c_2
2

Choose a method

Isolate a convenient variable for substitution, or match coefficients to eliminate one. If you need a particular method, say so in your question.

x−y=1⇒x=y+1x-y=1 \quad\Rightarrow\quad x=y+1
3

Solve and verify

Find every unknown, then substitute the values into each original equation. A solution must satisfy all of them.

(x,y)=(3,2)(x,y)=(3,2)

Choose an approach

Substitution, Elimination & Graphing

These are three ways to understand a two-variable linear system. If you want to see a particular algebraic method, include it in the question you enter above.

Substitution method solver`, text: `Isolate one variable, replace it in the other equation, solve the resulting one-variable equation, then back-substitute.

y=2x−1⇒3x+2(2x−1)=12y=2x-1 \quad\Rightarrow\quad 3x+2(2x-1)=12

Elimination method calculator`, text: `Multiply equations when needed to make one pair of coefficients opposites. Add the rows, solve, then back-substitute.

2x+3y=12and3x−3y=32x+3y=12 \quad\text{and}\quad 3x-3y=3

Solve by graphing

Rewrite two-variable linear equations as lines. Their intersection is the solution; parallel lines never meet, while coincident lines overlap.

y=mx+by=mx+b

Possible results

One Solution, No Solution or Infinitely Many

For two linear equations, the geometry of the lines tells you what kind of answer to expect.

1

One solution

Different slopes: the lines cross at one ordered pair.

y=x+1, y=−x+3⇒(1,2)y=x+1,\ y=-x+3 \quad\Rightarrow\quad (1,2)
2

No solution

Same slope, different intercepts: a contradiction appears.

0=50=5
3

Infinitely many

The equations describe the same line: an identity appears.

0=00=0

Three unknowns

Solving 3-Variable Systems of Equations

For a 3×3 linear system, write all three equations with the variables aligned. Eliminate the same variable from two pairs of equations to create a 2×2 system, solve it, and back-substitute to find the third value.

a1x+b1y+c1z=d1a_1x+b_1y+c_1z=d_1

Real-world systems

Systems of Equations in Word Problems

Ticket sales, mixtures, and rate problems can give two unknowns with two conditions. Define each unknown, write one equation per condition, then solve and check the result in the story.

a+s=200and10a+5s=1400a+s=200\quad\text{and}\quad 10a+5s=1400
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Worked examples

Systems of Equations Worked Step by Step

Elimination · two variables

2x+3y=12,x−y=12x+3y=12,\quad x-y=1
  1. 1.Multiply the second equation by 3: 3x−3y=33x-3y=3.
  2. 2.Add it to the first: 5x=155x=15, so x=3x=3.
  3. 3.Back-substitute into x−y=1x-y=1: 3−y=13-y=1, giving y=2y=2.
Answer:(x,y)=(3,2)(x,y)=(3,2)
Check:2(3)+3(2)=12and3−2=12(3)+3(2)=12\quad\text{and}\quad 3-2=1

Substitution · two variables

y=2x−1,3x+2y=12y=2x-1,\quad 3x+2y=12
  1. 1.Replace yy in the second equation: 3x+2(2x−1)=123x+2(2x-1)=12.
  2. 2.Simplify to 7x−2=127x-2=12, so x=2x=2.
  3. 3.Use the first equation: y=2(2)−1=3y=2(2)-1=3.
Answer:(x,y)=(2,3)(x,y)=(2,3)
Check:3=2(2)−1and3(2)+2(3)=123=2(2)-1\quad\text{and}\quad 3(2)+2(3)=12

Word problem · ticket sales

A theater sells 200 tickets. Adult tickets cost $10 and student tickets cost $5. Total revenue is $1,400. How many of each were sold?

  1. 1.Let aa count adults and ss count students. Then a+s=200a+s=200 and 10a+5s=140010a+5s=1400.
  2. 2.Substitute s=200−as=200-a: 10a+5(200−a)=140010a+5(200-a)=1400, so 5a=4005a=400.
  3. 3.Therefore a=80a=80 and s=200−80=120s=200-80=120.
Answer:80 adult tickets and 120 student tickets
Check:80+120=200and10(80)+5(120)=140080+120=200\quad\text{and}\quad 10(80)+5(120)=1400

Common Mistakes When Solving Systems

  1. 1

    Missing a negative sign

    Subtracting 2x−5y=42x-5y=4 means changing every sign, producing −2x+5y=−4-2x+5y=-4.
  2. 2

    Stopping after one variable

    The solution to a two-variable system is an ordered pair (x,y)(x,y), not just one coordinate.
  3. 3

    Substituting without parentheses

    When subtracting y=2x−1y=2x-1, write −(2x−1)=−2x+1-(2x-1)=-2x+1, not −2x−1-2x-1.
  4. 4

    Misreading a contradiction

    If all variables cancel and you get 0=80=8, the system has no solution; 0=00=0 needs further checking.

FAQ

Frequently Asked Questions

Math AI Solver lets you enter equations and follow a worked solution without signing up. Enter each equation on a separate line, and check the result in every original equation.

Substitution is convenient when a variable is already isolated or has a coefficient of 1 or -1. Elimination is useful when coefficients can be matched so one variable cancels when equations are added or subtracted.

It is an identity, not a single solution. For two linear equations, if one equation is a multiple of the other, the lines coincide and there are infinitely many solutions. Check the remaining equation before concluding that the whole system is dependent.

Yes. Put each equation on a new line and include all three variables. A common hand method eliminates one variable twice, solves the remaining two-variable system, and then back-substitutes to find the third variable.

Graph each equation on the same axes. An intersection is a shared solution, parallel distinct lines have no solution, and overlapping lines have infinitely many solutions.