We solve the system of linear equations:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Enter the coefficients into the fields below.
First equation
a₁
b₁
c₁
Second equation
a₂
b₂
c₂

Our linear combination calculator is here whenever you need to solve a system of equations using the linear combination method (also known as the elimination method). If you want to learn what the linear combination method is or how to use the linear combination method, check the article below.

We explain what combining linear equations means and how to use the linear combination method to solve systems of linear equations. Most importantly, we show you several very detailed step-by-step examples of systems solved with the linear combination method.

What is a system of linear equations?

For an equation to be linear, all its variables must be in the first power: they cannot be squared/cubed, nor under a root, nor placed in the denominator. If some numbers satisfy several linear equations at once, we say that these numbers are a solution to the system of those linear equations.

In school, we most often encounter systems of two linear equations in two variables. In general, such a system takes the form:

a1x + b1y = c1

a2x + b2y = c2

where:

  • x and y are the variables;
  • a1, b1, c1 are the coefficients of the first equation; and
  • a2, b2, c2 are the coefficients of the second equation.

What is the linear combination method?

The linear combination of equations is a method for solving systems of linear equations. The key idea is to combine the equations into a system of fewer and simpler equations. If we deal with two linear equations in two variables, we want to combine these equations into one equation with a single variable.

By combining linear equations we mean multiplying one or both equations by suitably chosen numbers and then adding the equations together. Multipliers must allow the operation of addition to cause one of the variables to vanish. We then end up with a single equation in one variable, which we can easily solve. Once we have solved this equation, we substitute the value we've found into one of the original equations. This leads to another equation in one variable, which we quickly solve. This way, we've solved the system using linear combination!

Are there other methods for solving systems of linear equations? Sure! You have probably heard about the substitution method. There is also the Gaussian elimination method and Cramer's rule.

How to use this linear combination calculator?

If you want to quickly solve a system of equations using linear combination, our tool is the best choice! To use it, follow the steps below:

  1. Tell us the coefficients of your system: type them in the appropriate fields.
  2. Our linear combination calculator solves the system using the linear combination method...
  3. ...and displays and explains all the intermediate steps!
  4. Go to the advanced mode of the linear combination calculator if you want to compute the solution with higher precision (with a higher number of sig figs). By default, our linear combination calculator uses six sig figs.

How to use the linear combination method?

For the system of equations:

a1x + b1y = c1

a2x + b2y = c2

we can always use the least common multiple of a1 and a2. Namely, put:

m1 := LCM(a1, a2) / a1

m2 := LCM(a1, a2) / a2

and multiply the first equation by m1 and the second equation by -m2. This leads to the following system:

LCM(a1, a2)x + [LCM(a1, a2)b1/a1]y = LCM(a1,a2)c1/a1

-LCM(a1, a2)x - [LCM(a1, a2)b2/a2]y = -LCM(a1, a2)c2/a2

We have created opposite coefficients for the variable x! All we need to do is to add these equations together: we will obtain an equation with the y variable only.

In some particular situations, this LCM approach boils down to elementary operations:

  • When the coefficients of one variable are opposite numbers, both multipliers are equal to 1.

  • When the coefficients of one variable are equal, one multiplier is equal to 1 and the other to -1.

It may sometimes happen that you eliminate both variables at once. First of all, do not panic. You arrived at a statement about numbers. Depending on whether the statement you got is true, like:

0 = 0 or 7 = 7,

or false, e.g.:

0 = 1 or 5 = 7,

draw conclusions about the system:

  • If the statement is false, then the system has no solution.

  • If the final statement is true, then the system has infinitely many solutions.

Linear combination method examples

Wow, we've learned a lot about what the linear combination method is! It's time to solve a few systems of linear equations using linear combinations.

  1. Consider the system of equations:

    x - 4y = 1

    -2x + 4y = 2

    and solve it using linear combination.

    • Add the two equations together:

      -x = 3

    • Solve for x:

      x = -3

    • Substitute x = -3 into the first equation:

      -3 - 4y = 1

    • Solve for y:

      -4y = 4

      y = -1

    • Solution: x = -3, y = -1

  2. Solve using linear combination:

    2x + 3y = 3

    2x - y = -3

    • First, multiply the first equation by -1:

      -2x - 3y = -3

      2x - y = -3

    • Add the equations, which results in eliminating x:

      -4y = -6

    • Solve for y:

      y = 1.5

    • Substitute y = 1.5 into the second equation:

      2x - 1.5 = -3

    • Solve for x:

      2x = -1.5

      x = -0.75

    • Solution: x = -1.5, y = -0.75

  3. Solve the system using linear combination:

    3x - 7y = 1

    4x + 4y = -2

    • Use the LCM approach: find the calculate the least common multiplicity of 3 and 4:

      LCM(3, 4) = 12.

      The multipliers are:

      m1 = 4 and m2 = -4 and the system is:

      12x - 28y = 4

      -12x - 12y = 6

    • Add the equations:

      -40y = 10

    • We solve for y:

      y = -0.25

    • We substitute y = -0.25 into the second equation:

      4x - 4 ⋅ (-0.25) = 0

    • We solve for x:

      4x = -1

      x = -0.25

    • Solution: x = -0.25, y = -0.25

Anna Szczepanek, PhD