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Solving Linear Systems by Substitution

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Solving Linear Systems by Substitution

In algebra, a system of linear equations is a set of two or more linear equations with the same variables. A solution to a linear system is a set of values for the variables that satisfies all of the equations in the system. One way to solve a linear system is by using the substitution method.

What is the Substitution Method?

The substitution method is a technique for solving a linear system of equations. It involves solving one equation for one variable in terms of the other variable and then substituting that expression into the other equation. This process eliminates one variable, allowing you to solve for the remaining variable. Once you have solved for one variable, you can substitute that value back into either of the original equations to find the value of the other variable.

Steps for Solving a Linear System by Substitution

Here are the steps for solving a linear system of two equations with two unknowns using the substitution method:

  1. Solve one equation for one variable in terms of the other variable. For example, if you have the equations:
    • 2x + y = 5
    • x – 3y = 1

    You could solve the second equation for x:

    x = 3y + 1

  2. Substitute the expression you just found into the other equation. In our example, we would substitute 3y + 1 for x in the first equation:

    2(3y + 1) + y = 5

  3. Solve the resulting equation for the remaining variable. In this case, we would solve for y:

    6y + 2 + y = 5

    7y + 2 = 5

    7y = 3

    y = 3/7

  4. Substitute the value you just found back into either of the original equations to solve for the other variable. We can substitute y = 3/7 into the equation x = 3y + 1:

    x = 3(3/7) + 1

    x = 9/7 + 1

    x = 16/7

  5. Write the solution as an ordered pair. The solution to the system of equations is (x, y) = (16/7, 3/7).

Example

Let’s solve the following system of equations using the substitution method:

  • 3x + 2y = 11
  • x – y = 2
  1. Solve the second equation for x:

    2. x = y + 2

  2. Substitute the expression for x into the first equation:

    3. 3(y + 2) + 2y = 11

  3. Solve for y:

    4. 3y + 6 + 2y = 11

    5y + 6 = 11

    5y = 5

    y = 1

  4. Substitute y = 1 into the equation x = y + 2:

    5. x = 1 + 2

    x = 3

  5. The solution to the system of equations is (x, y) = (3, 1).

Additional Resources

For further learning about solving linear systems by substitution, you can refer to the following resources:

The substitution method is a powerful tool for solving linear systems. It is a relatively straightforward method that can be used to solve a wide variety of problems.

Algebra