System of linear equations, also called simultaneous equations, can also be solved using the substitution method. This lesson will show how to solve a pair of linear equations with two unknown variables.
ax + by = c
dx + ey = f
Before you read this lesson, make sure you understand how to solve linear equations.
Example #1: Solve the following system using the substitution method
x + y = 20
x − y = 10
Step 1
You have two equations. Pick either the top equation or the bottom equation and solve for either x or y.
Since I am the one solving it, I have decided to choose the equation at the bottom (x − y = 10) and I will solve for x.
x − y = 10
Add y to both sides
x − y + y = 10 + y
x = 10 + y
Step 2
Since you used the equation at the bottom and solved the equation for the variable x, you will substitute x into the equation on top (x + y = 20)
Using x + y = 20, erase x and write 10 + y since x = 10 + y
We get 10 + y + y = 20
Step 3
Solve for y
10 + y + y = 20
10 + 2y = 20
Minus 10 from both sides
10 − 10 + 2y = 20 − 10
2y = 10
Divide both sides by 2
y = 5
Step 4
Since you have found the value of the variable y, you can plug its value into either the top equation or the bottom to get x.
Replacing y into x + y = 20 gives x + 5 = 20
Minus 5 from both sides
x + 5 − 5 = 20 − 5
x = 15
The solution to the system is x = 15 and y = 5
Indeed 15 + 5 = 20 and 15 − 5 = 10
Example #2: Solve the following system using the substitution method
3x + y = 10
-4x − 2y = 2
Step 1
You have two equations. Pick either the first equation (top) or the second equation (bottom) and solve for either x or y.
I have decided to choose the equation on top (3x + y = 10) and I will solve for y.
3x + y = 10
Subtract 3x from both sides
3x − 3x + y = 10 − 3x
y = 10 − 3x
Step 2
Since you used the equation on top to solve for y, you will substitute y into the equation at the bottom (-4x − 2y = 2)
Using -4x − 2y = 2, erase y and write 10 − 3x keeping in mind that there is a multiplication between 2 and y.
We get -4x − 2×(10 − 3x ) = 2
Step 3
Solve for x
-4x − 2 ×(10 − 3x ) = 2
-4x − 20 + 6x = 2 (After multiplying -2 by 10 and -2 by -3x)
2x − 20 = 2
Add 20 to both sides
2x − 20 + 20 = 2 + 20
2x = 22
Divide both sides by 2
x = 11
Step 4
Now you have x, you can replace its value into either the first equation on top or the second equation at the bottom to get y.
Replacing x into 3x + y = 10 gives 3 × 11 + y = 10
33 + y = 10
Minus 33 from both sides
33 − 33 + y = 10 − 33
y = -23
The solution to the system is x = 11 and y = -23
Indeed,
3 × 11 + -23 = 33 + -23 = 10 and -4 × 11 − 2 × -23 = -44 + 46 = 2
You should have noticed so far that the reason we call this method the substitution method is because after you have solve for a variable in one equation,you substitute the value of that variable into the other equation.
Example #3: Solve the following system using the substitution method
2x + y = 8
2x + y = 8
Step 1
Pick the equation on top and solve for y.
2x + y = 8
2x - 2x + y = 8 - 2x
y = 8 - 2x
Step 2
Substitute the value of y in the equation at the bottom.
2x + 8 - 2x = 8
8 = 8
Notice that instead of finding a value for x, we end up with a true statement (8 = 8)
When this happens, it means that the system has an infinite number of solutions. Any ordered pair (x,y) that solves the system is a solution. Find ordered pairs by choosing anything you want for x and then solve for y.
For example, if I choose 1 for x, then 2(1) + y = 8
2 + y = 8
y = 6
A solution is (1, 6). You can find an infinite amount if you keep choosing arbitrary value for x.
If you solve a system by substitution and you end up with a true statement, it means that the system has an infinite number of solutions.
Example #4: Solve the following system using the substitution method
2x + y = 4
2x + y = 8
Step 1
Pick the equation on top and solve for y.
2x + y = 4
2x - 2x + y = 4 - 2x
y = 4 - 2x
Step 2
Substitute the value of y in the equation at the bottom.
2x + 4 - 2x = 8
4 = 8
Notice that instead of finding a value for x, we end up with a false statement (4 = 8)
If you solve a system by substitution and you end up with a false statement, it means that the system has no solution.
Jan 26, 23 11:44 AM
Jan 25, 23 05:54 AM