Substitution

To solve a system of equations in a more systematic way, you use substitution. Toward this end, you first solve one of the equations in the system for one of its variables. You then substitute the solution into the other equation.

Here is a system of equations that you can solve in this way:

x + y = 12

3x + y = 4

To solve this set of equations through substitution, you can solve either equation for either variable. In this instance, begin by solving the first equation for y:

x + y = 12

y = 12 - x

You can then proceed to substitute the expression 12 - x into the second of the equations. To do so, you proceed as follows:

Having arrived at this solution for x, you can then return to either of the equations in the system and solve for y. Accordingly, if you use the first equation,

you proceed as follows:

-4 + y = 12

y = 12 + 4

y = 16

Given the solution to y, you then have at hand the ordered pair (-4, 16). You can test the validity of this ordered pair by substituting its two values into the equations:

-4 + 16 = 12

12 = 12

Likewise, with the second equation,

16 = 12 - (-4)

16 = 16

Exercise Set 11.1 Use substitution to solve these systems of equations for the value of x and y. Check your answers. 6x - 2y = -4 3x + 4y = 1 4x + 6y = -6 -4x + 4y = 16 x - 6y = 4 2x - 2y = 7 2x + y = 6 3x + 4y = 4