Solving Systems of Equations By Substitution
After studying this lesson, you will be able to:
 Solve systems of equations by substitution.
To Solve a System of Equations by Substitution:
1. Solve either equation for a variable. (Hint: Try to solve
for a variable that does not have a coefficient.)
2. Substitute on equation into the other equation. This will
eliminate one variable.
3. Solve for the remaining variable.
4. Substitute the solution into the other equation and solve
for the other variable.
Example 1
Solve x + y = 6, x = y + 2
1 st : The second equation is already solved for a variable so
we don't have to do anything in this step.
2 nd : Now we substitute x = y + 2 into the first equation.
Since we know that x is the same as y + 2, we can substitute y +
2 for x in the first equation:
( y + 2 ) + y = 6
3 rd : Solve for y which is the remaining variable:
2y + 2 = 6
2y = 4
y = 2
4 th : Now we substitute the solution ( y = 2 ) into the
second equation:
x = y + 2
x = (2) + 2
x = 4
The solution is (4, 2)
Example 2
Solve 2x + y = 13, 4x 3y = 11
1 st : We need to solve one equation for a variable. Let's
solve the first equation for y since that is the variable without
a coefficient:
2x + y = 13
y = 13  2x
2 nd : Now we substitute y = 13  2x into the second equation.
Since we know that y is the same as 13  2x, we can substitute 13
 2x for y in the second equation:
4x  3y = 11
4x  3 (13  2x ) = 11 (making the substitution)
3 rd : Solve for x which is the remaining variable:
4x  3 (13  2x ) = 11
4x  39 + 6x = 11
10x 39 = 11
10x = 50
x = 5
4 th : Now we substitute the solution ( x = 5 ) into the first
equation:
2x + y = 13
2 (5) + y = 13
10 + y = 13
y = 3
The solution is (5, 3)
Special Cases: If you end up with a system
where all variables cancel out, you have what we might call a
"special case".
If the statement you're left with is true, the solution will
be
If the statement you're left with is false, the solution will
be Ã˜
Example 3
Solve x  3y = 6, x  3y = 6
1 st : We need to solve one equation for a variable. Let's
solve the first equation for x since that is the variable without
a coefficient:
x  3y = 6
x = 6 + 3y
2 nd : Now we substitute x = 6 + 3y into the second equation.
x  3y = 6
(6 + 3y) 3y = 6 (making the substitution)
3 rd : Solve for y which is the remaining variable:
(6 + 3y) 3y = 6
6 = 6 All variables cancelled out so we have a special case.
Since the remaining statement is false, the solution is
Example 4
Solve x + 2y = 5, 3x  15 = 6y
1 st : We need to solve one equation for a variable. Let's
solve the first equation for x since that is the variable without
a coefficient:
x + 2y = 5
x = 5  2y
2 nd : Now we substitute x = 5  2y into the second equation.
3x  15 = 6y
3 (5  2y)  15 = 6y (making the substitution)
3 rd : Solve for y which is the remaining variable:
3 (5  2y)  15 = 6y
15  6y 15 = 6y
6y = 6y
6 = 6 All variables cancelled out so we have a special case.
Since the remaining statement is true, the solution is Ã˜
