Solving by Substitution
Learning Goals
By the end of this lesson I will be able to:
What is Substitution?
Remember, the idea of substitution is simply replacing one unknown (say x) with another value.
For example:
Substitute x = 5 into 3x - 8y = 55:
3(5) - 8y = 55 Replace all x’s with 5
15 - 8y = 55 Multiply 3 by 5
-8y = 40 Subtract 15 from both sides
y = -5 Divide both sides by -8
What does it Do for us?
In the previous example, substituting x = 5 allowed us to solve for y, in one specific case.
That is our goal in our systems of equations: Solve a system of linear equations.
Let’s solve the following using substitution:
x - 2y = -6
y = 3x - 7
How do we use Substitution?
x - 2y = -6
First we have to figure out WHAT we can substitute. In order to do this, we need to have one variable isolated (by itself).
Our second equation, y = 3x + 7 has an isolated y! Let’s use that.
y = 3x - 7
x - 2(3x - 7)= -6
How do we use Substitution? Continued
Now we distribute the -2 over 3x - 7.
x - 2(3x - 7)= -6
x -6x + 14 = -6
-2 times 3x = -6x
-2 times -7 = 14
Gather like terms...
-5x + 14 = -6
Subtract 14 from both sides...
-5x = -20
Divide by -5...
x = 4
Now what?
So now we know that x = 4… Are we done?
Nope! This is only half of our point of intersection. We need a y value.
Now we substitute x = 4 into either of our original equations.
x - 2y = -6
y = 3x - 7
4 - 2y = -6
-2y = -10
y = 5
y = 3(4) - 7
y = 12 - 7
y = 5
(It doesn’t matter which one we pick!)
And LAst But Not Least...
We now need to check that (4, 5) is the right point:
x - 2y = -6
y = 3x - 7
L.S. = (4) - 2(5)
= 4 - 10
= -6 = R.S.
R.S. = 3(4) - 7
= 12 - 7
= 5
L.S. = y = 5 = R.S.
In both cases, L.S. = R.S. so we MUST have the right answer.
Example 2
Solve by substitution:
3x + 4y = -16
y = ¼ x
Example 3
Solve by substitution:
y = 12x - 15
y = x - 4
Example 4
Solve by substitution:
7x - 3y = 6
2x + 5y = 31
Example 5
Solve by Substitution:
6x - 3y = 16
-3x + y = 12