Solving Systems of Equations by Substitution

To solve a linear system of equations by the method of substitution:

1) first solve for one of the variables in one of the equations … it does not matter if you solve for x or y … check both equations and solve for the variable that has the smallest coefficient,
2)  substitute the value of the variable that you solved for in step one into the other equation … the two equations must interact with one another,
3)  step 2 will result in an equation in one variable, solve this equation for the variable,
4)  use the value found in step 3 to find the value of the remaining variable, you may use either equation to find the second value,
5)  check the solution in both equations to be sure that the solution works in both equations.

Solve the system of equations … 3x – 2y  = 19 and x + y = 8 … by substitution.

First, exam both of the equations and decide for which variable to solve.  Since the second equation has both an x term and a y term with coefficients of 1, I can choose to solve for either x or y in the second equation.  I choose to solve for x in the second equation.  Solving for x, I find that x = 8 – y.  Now, I use the value for x, which is 8 – y, and substitute that value into the first equation for the value of x.

     3(8 – y) – 2y = 19 … now solve for y

     24 – 3y – 2y = 19

     – 5y = 19 – 24

     – 5y =  – 5

   y  =  1  … now use this value and substitute it into either of the original equation to find the value of x.

     3x – 2y = 19

     3x – 2(1) = 19

     3x = 19 + 2

     3x = 21

     x = 7 … the solution set is the point ( 7, 1 )

Check:  the solution set should work in both of the equations 3x – 2y  = 19 and x + y = 8.
Since 3(7) – 2(1)  = 19 and (7) + (1) = 8, then the point ( 7, 1) is the solution to this system of equations.