5 Solving Systems of Equations
Finding the value of more than one variable in more that one equation is called “solving a system of equations”. There are four methods to solve a system of equations:
- Addition
- Subtraction
- Multiplication
- Substitution
Subtraction Method:
This method is best if you see that both equations have one variable with the same coefficient with the same sign.
- Write one equation above another
- Match up the x and y variables and the whole numbers.
- Write the subtraction sign outside the parentheses on the bottom equation
2x + 4y = 8
-(2x + 2y = 2)
4. Subtract like terms
2x – 2x = 0
4y – 2y = 2y
8 – 2 = 6
2x + 4y = 8 -(2x + 2y = 2) = 0 + 2y = 6
5. Solve for the remaining term
2y = 6
Divide 2y and 6 by 2 to get y = 3
6. Plug the term you found in 5 back into one of the equations from the original set to find the value of the first term.
Plug y = 3 into the equation 2x + 2y = 2 and solve for x.
2x + 2(3) = 2
2x + 6 = 2
2x = -4
x = – 2
You have solved the system of equations by subtraction. (x, y) = (-2, 3)
Addition Method:
This method is best if you see that both equations have one variable with the same coefficient with the opposite sign.
- Write one equation above another
- Match up the x and y variables and the whole numbers.
- Write the addition sign outside the parentheses on the bottom equation
3x + 6y = 8
+(x – 6y = 4)
4. Add like terms
3x + x = 4x
6y + -6y = 0
8 + 4 = 12
+(x – 6y = 4)
= 4x + 0 = 12
5. Solve for the remaining term
4x + 0 = 12
4x = 12
Divide 4x and 12 by 3 to get x = 3
6. Plug the term you found in 5 back into one of the equations from the original set to find the value of the first term.
Plug x = 3 into the equation x – 6y = 4 to solve for y.
3 – 6y = 4
-6y = 1
Divide -6y and 1 by -6 to get y = -1/6
You have solved the system of equations by addition. (x, y) = (3, -1/6)
Multiplication Method:
When you use the multiplication method, none of the variables will have matching coefficients
1. Write one equation above another
3x + 2y = 10
2x – y = 2
Now, multiply one or both of the equations by a number that would make one of the variables have the same coefficient. In this case, you can multiply the entire second equation by 2 so that the variable -y becomes -2y and is equal to the first y coefficient. Here’s how to do it:
2.Multiply one or both equations until one of the variables of both terms have equal coefficients.
2 (2x – y = 2)
4x – 2y = 4
3.Now, just use the addition or subtraction method on the two equations based on which method would eliminate the variable with the same coefficient.
3x + 2y = 10
+ 4x – 2y = 4
7x + 0 = 14
7x = 14
4.Solve for the remaining term
7x = 14,
x = 2.
5.Plug the term you found in 4 back into one of the equations from the original set to find the value of the first term.
x = 2 —> 2x – y = 2
4 – y = 2
-y = -2
y = 2
You have solved the system of equations by multiplication. (x, y) = (2, 2)
Substitution Method:
The substitution method can be used for any equations.
1.Solve one of the equations for either x= or y=
2x + 2y = 3
x-4y = -1
x = 4y – 1
2.Substitute the solution from step 1 into the other equation.
2(4y-1) + 2y = 3
3.Solve this new equation:
8y-2+2y=3
10y-2=3
10y=5
y=.5
4.Put this answer back in the original equation to get the other variable:
x = 4(.5) – 1
x = 2 – 1
x = 1
