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

This is Omni's substitution method calculator, which helps you solve a system of equations by the substitution method. What is the substitution method?, you may ask. And how to use the substitution method?

Keep reading! We will give the definition of the substitution method, enumerate the substitution method steps, and take you through several examples of systems solved with the substitution method step by step (with answers!). We will also explain how to do the substitution method when a system turns out to be consistent or dependent.

What are systems of linear equations?

We say that we want to solve a system of linear equations if we have several equations and want to find numbers that solve all these equations simultaneously.

Let's recall that an equation is linear if all the variables in that equation are in the first power. This means that they cannot be squared, nor cubed, nor put under any root (in particular under a square root or cube root). They also cannot be denominators of fractions.

This substitution method calculator works for systems of two linear equations in two variables. These are the systems most common encountered in homework! 😉 They take the following 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 substitution method?

The substitution method is a method of solving systems of linear equations. The main idea behind solving systems with the substitution method is to choose one of the equations, solve it for one of the variables, and plug the result into the other equation. This way, we obtain an equation with one variable, which we can easily solve. Once we have found the value of one variable, we utilize it to find the other variable. This is how we use the substitution method to solve the system of equations. Go to the next section to learn more about the substitution method steps.

Do you know that there are other methods for solving systems of linear equations? You're not forced to solve by substitution method! You can use the elimination method (also known as the linear combination method) or Gaussian elimination, or, once you've mastered matrix determinants, Cramer's rule.

How to use our substitution method calculator?

This system of equation substitution method tools is easy to use:

  1. Enter the coefficients of the system of linear equations in their respective fields.
  2. The complete solution by substitution method appears below the substitution method calculator!
  3. All the substitution method steps are here in case you need them.
  4. If you need the system solved with a higher precision (number of significant figures) than the six figs that this substitution method calculator uses by default, simply click the advanced mode button and set your desired precision.

Substitution method step-by-step

We have already explained what the substitution method is about and what the main idea behind it is. Now, let's discuss in detail how to do the substitution method:

  1. Choose one of the equations.

  2. In the chosen equation, choose one of the variables.

  3. Solve this equation for this variable.

  4. Plug the result into the other equation (the one you didn't choose in Step 1). This is the essence of solving systems by the substitution method!

  5. You have obtained a one-variable equation - solve it!

  6. Substitute the value you got in Step 5 into one of the original equations.

  7. Solve this new one-variable equation.

  8. That's it! You solved the system of equations by the substitution method. If you want, you may test your solution: substitute the values you obtained into the system and see if everything is OK.

⚠️ It may sometimes happen that you try to solve a system and suddenly both variables vanish 😱 Keep calm! The variables are no more, but you arrived at some statement about numbers. All you need to do is draw conclusions about the system depending on whether the statement you got is true (like 0 = 0 or 17 = 17) or false (e.g., 0 = 1 or 15 = 17):
  • If the statement is false, then the system is inconsistent, which means it has no solution.

  • If the statement is true, then the system is dependent, i.e., it has infinitely many solutions.


Substitution method examples with answers

In this section, we show you step by step how to solve several systems using the substitution method so that you can see how to do the substitution method in practice.

  1. Use the substitution method to solve the system of equations:

    3x - 4y = 6

    -x + 4y = 2

    • Solve the second equation for x:

      x = 4y - 2

    • Substitute 4y - 2 for x into the first equation:

      3(4y - 2) - 4y = 6

    • Solve the above equation for y:

      12y - 6 - 4y = 6

      8y = 12

      y = 1.5

    • Substitute y = 1.5 into the second equation:

      -x + 4 * 1.5 = 2

      -x + 6 = 2

    • Solve for x:

      x = 6 - 2

      x = 4

    • Solution: x = 4, y = 1.5

    • Test the solution:

      3 * 4 - 4 * 1.5 = 12 - 6 = 6

      We see that the first equation is OK.

      -4 + 4 * 1.5 = -4 + 6 = 2

      And the second equation is OK as well.

  2. Solve using substitution method:

    2x + 3y = 5

    2x + 7y = -3

    • Solve the first equation for x:

      x = -1.5y + 2.5

    • Substitute x = -1.5y + 2.5 into the second equation:

      2*(-1.5y + 2.5) + 7y = -3

    • Solve the above equation for y:

      -3y + 5 + 7y = -3

      4y = -8

      y = -2

    • Substitute y = -2 into the first equation:

      2x + 3*(-2) = 5

    • Solve for x:

      2x = 11

      x = 5.5

    • Solution: x = 5.5, y = -2

  3. Now, we will see how to solve the following system of linear equations using the substitution method:

    6x - 3y = 12

    2x - y = 4

    • Solve the second equation for y:

      y = 2x - 4

    • Substitute y = 2x - 4 into the first equation:

      6x - 3*(2x - 4) = 12

    • Solve the above equation for x:

      6x - 6x - 12 = 12

      0 = 0

    • We eliminated both variables and arrived at a true statement. Hence, this system of equations has infinitely many solutions!

Anna Szczepanek, PhD