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Solving Linear Systems Using Substitution

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Solving Linear Systems Using Substitution

In algebra, a system of linear equations is a set of two or more linear equations with the same variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. One common method for solving linear systems is the substitution method.

What is the Substitution Method?

The substitution method involves solving one equation for one variable in terms of the other variable and then substituting that expression into the other equation. This results in an equation with only one variable, which can then be solved. Once the value of one variable is known, it can be substituted back into either of the original equations to solve for the other variable.

Steps to Solve a System of Linear Equations Using Substitution

  1. Solve one equation for one variable in terms of the other variable. Choose the equation that is easiest to solve for one variable. For example, if one equation has a variable with a coefficient of 1, it is easier to solve for that variable.
  2. Substitute the expression from step 1 into the other equation. This will result in an equation with only one variable.
  3. Solve the equation from step 2 for the remaining variable.
  4. Substitute the value of the variable from step 3 back into either of the original equations to solve for the other variable.
  5. Check your solution by substituting the values of both variables into both original equations. The solution should satisfy both equations.

Example

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

Equation 1: 2x + y = 5

Equation 2: x - 3y = 1

  1. Solve Equation 2 for x: x = 3y + 1
  2. Substitute the expression for x into Equation 1: 2(3y + 1) + y = 5
  3. Solve for y: 6y + 2 + y = 5
    7y + 2 = 5
    7y = 3
    y = 3/7
  4. Substitute the value of y back into Equation 2: x - 3(3/7) = 1
  5. Solve for x: x - 9/7 = 1
    x = 16/7
  6. Check the solution:
    2(16/7) + 3/7 = 5 (satisfies Equation 1)
    16/7 - 3(3/7) = 1 (satisfies Equation 2)

Therefore, the solution to the system of equations is x = 16/7 and y = 3/7.

Additional Resources

For further learning and practice, you can explore the following resources:

Linear Equations