General Educational Development (GED): Systems of Equations

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Solving Systems of Equations

A system of equations is a collection of two or more equations with a same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system.

Step 2: Substitute the value of  Y in one of the equations with the other equation. This will change equation to an equation with just one variable.
Step 3: Solve for the X variable.
Step 4: Substitute the answer with any of the original equations and solve for the Y variable.

Step 5: Check your answers by substituting the values of x and y in each of the original equations. If, after the substitution, the left side of the equation equals the right side of the equation, you know that your answers are correct.

Example:

Equation A:  y=2x+4

Equation B:   y=3x+2

The first step is to take one of the equations and substitute it into the other equation for y variable. So we will take the first equation and put it in the second equation in place of the y.

3x + 2 = 2x + 4

next step is to get the x by itself so we will minus 2x on each side of the equation

3x - 2x + 2 = 2x -2x + 4

now we add the like terms

x + 2 = 4

next step is to solve for x

we will minus 2 on both sides of the equation

x + 2 - 2 = 4 - 2

Now we will need to minus 14x from each side of equation

now we will have

x = 2

the next step is to take our answer and add it into one of the original equations to solve for y.

now we will take the second equation and replace the x with the number 2.

y = 3(2) + 2

y = 6 +2

y = 8

 

the Solution is (2, 8)

verify that both answers solve the equations with both sides equaling the same in each of the original equations

8 = 2(2) +4                                       8 =3(2) +2

8 = 4 +4                                             8 = 6 + 2

8 = 8                                                   8 = 8

Graphing

Make sure both equations are in the slope intercept form before graphing:  y = mx + b

The coefficient in front of the x value will be your slope and the other number will be the y-intercept.

Example: y = x +3 

Slope will be 1 and are y-intercept will be 3

So when you have a system of equations then you can graph both lines. the place where they intercept is your solution.

 

Having a negative slope, the line will decrease or fall from left to right

The slope is positive thus the line will increase or rise from left to right

 

Plotting Points 

Worksheet on plotting points on a graph

The Elimination Method for solving systems of linear equations uses the addition property of equality or the multiplication property of equality.

Addition property: states that if you add the same number to both sides of an equation, the sides remain equal

 

Multiplication Property:  states that when we multiply both sides of an equation by the same number, the two sides remain equal.

 

Addition property of equality

The first step to solving by elimination is to add the equations to Eliminate one of the variables.

Now you solve for the remaining variable.

Substitute the answer you got for the remaining variable into one of the equations to solve for the other variable

The answers should be written: solution = (x , y) format.

 

Example:

3y+2x=65y−2x=10

First step is to add the two equations together to eliminate one of the variables

3y + 2x = 6

5y -2x = 10

8y = 16

We were able to eliminate the x variable. The next step is to solve for y. To do this we would divide both sides of the equation by 8.

we will get an answer of y = 2

The next step is to take are y = 2 and substitute it into one of the equations to find the value of X. In this case we will choose the first original equation to solve for x. The Y in the equation will be changed to an 2.

3(2) + 2x = 6

6 + 2x = 6

solve for x we would subtract six from each side of the equation which will leave us with

2x = 0

Divide both sides by 2 and we get 

X = 0

the solution to these equations will be:

Solution = ( 0, 2)

Plug these numbers into each of the original equations to make sure they are correct

 

 

You can also solve the equations by subtracting the opposite of one of the equations with the other if the variables can not be eliminating by adding

Example:

2x + y = 12

 −3x + y = 2.

The first thing we will do is determine that if we add these together it will not eliminate a variable so what we will need to do now is take one of the equations and add the opposite of the other equation.

2x + y = 12

+ 3x - y = -2

5x = 10

This will now eliminate the y variable and now we can solve for the X value

5x = 10

5        5

x = 2

Now we take are answer and substitute it into one of the original equations to solve for y.

2(2) + y = 12 

4 + y = 12

Minus 4 on each side of the equation

y = 8

 

Multiplication property of equality

If you have an issue where if you add the two equations together and they don't eliminate a variable then you would use this method. 

You can begin by multiplying one or both of the equations with a constant to obtain an equivalent linear system where you can eliminate one of the variables by addition or subtraction.

Then solve for the remaining variable

Substitute answer into one of the equations for that variable and then solve to find the value of the remaining variable.

 

Example:

3x+y=9

5x+4y=22

begin by Multiplying the first equation by -4 to get a linear equation where we can eliminate one of the variables

-4(3x +y) = 9 x -4

-12x - 4y = -36

Now you want to add the new equation with the bottom equation

-12x - 4y = -36

+ 5x + 4y = 22

-7x = -14

Now we will continue by solving for x

-7x = -14

                                                          -7       -7

x = 2

The next step is to plug in the X value into one of the equations and solve for y

5x+4y=22

5(2) + 4y = 22

10 + 4y = 22

10-10 +4y = 22 - 10

4y = 12

4        4

y = 3

Solution is (2, 3)

verify that the solutions are correct by putting them into the original equations