Solving systems of equations

SOLVING SYSTEMS OF EQUATIONS

March 31, 2011 Samantha Billingsley

Solving systems of equations

You can solve a system of equations with algebra as long as there are at least the same number of equations as variables (for two variables, you need two equations, etc.).

There are three ways to solve systems of equations algebraically: addition, subtraction, substitution.

Solving by addition

We solve equations by addition when both equations contain the same term, but with opposite signs.

For example: 4x + 5y = 81       - 2x - 5y = - 63

Notice that we have the term +5y in the first equation and -5y in the second equation.

Solving by addition

To solve this system, we simply add the two equations together like this:

4x + 5y = 81

+ (-2x -5y = -63)

2x = 18

Because +5y and -5y cancel out, we are left with only one variable and can easily solve the equation.

2x / 2 = 18/2

x = 9

Solving by addition

Now that we know that x = 9, we plug this value for x into one of our original equations:

4x + 5y = 81

4(9) + 5y = 81

36 + 5y = 81

36 - 36 + 5y = 81-36

5y = 45

5y/5 = 45/5

y = 9

Solving by addition

Our answer is x = 9, y=9. We can check this by plugging both values into the second equation:

-2x - 5y = -63

-2(9) - 5(9) = -63

-18 - 45 = -63

-63 = -63

Practice

Now try solving this system by addition:

2x + 3y = - 5

5x - 3y = 61

Solution

2x + 3y = -5

+ (5x - 3y = 61)

7x = 56

7x / 7 = 56 / 7

x = 8

2(8) + 3y = -5

16 + 3y = -5

16 -16 + 3y = -5 - 16

3y = -21

3y / 3 = -21 / 3

y = -7

Solving by subtraction

We can solve systems by subtraction when both equations contain the same term with the same sign.

For example:

-4x - y = 11

   -4x - 2y = 10

Notice that we have the term -4x in both equations.

Solving by subtraction

To solve, subtract the second equation from the first:

-4x - y = 11- (-4x - 2y = 10)

y = 1

Notice that both -4x’s cancel out when we subtract the two equations. We are left

with just one variable, y.

Solving by Subtraction

Now, plug this value for y into one of the equations.

-4x - y = 11

-4x -1 = 11

-4x -1+1 = 11+1

-4x = 12

-4x / -4 = 12 / -4

x = -3

Solving by Subtraction

We have x = -3, y = 1. We can check this by plugging both values into the second equation.

-4x - 2y = 10

-4(-3) - 2(1) =10

12 - 2 = 10

10 = 10

Practice

Try solving the following system by subtraction:

x + 4y = 21

x - 3y = - 28

Solution

x + 4y = 21- (x - 3y = - 28)

7y = 49

7y / 7 = 49/ 7

y = 7

x + 4(7) = 21

x + 28 = 21

x + 28 - 28 = 21 - 28

x = -7

Solving by substitution

If you cannot solve by addition or subtraction, you must solve by substitution.

Take this system for example:

x + 5y = 34

2x + 4y = 26

First, solve one equation for one variable (leaving it in terms of the other variable). In this case, we will solve the first equation for x, in terms of y.

x + 5y - 5y = 34 - 5y

x = 34 - 5y

Solving by substitution

Next, substitute your solution into the second equation.

x = 34 - 5y

2x + 4y = 26

2(34 - 5y) + 4y = 26

Using the distributive property, we get:

68 -10y + 4y = 26

68 -6y = 26

68 - 68 -6y = 26 – 68

-6y = -42

-6y / -6 = -42 / -6

y = 7

Solving by substitution

Now that we know y = 7, we can plug this value into our previous solution for x.

x = 34 - 5y

x = 34 - 5(7)

x = 34 - 35

x = -1

Solving by substitution

We have x = -1, y = 7. To check, plug both values into one of the original equations.

2x + 4y = 26

2(-1) + 4(7) = 26

-2 + 28 = 26

26 = 26

Practice

Try solving the following system by substitution:

y = - x + 3

- 5x = - 43 + y

Solution

y = - x + 3

- 5x = - 43 + y

-5x = - 43 + (-x + 3)

-5x = -43 - x + 3

-5x = -40 - x

-5x + x = -40 -x + x

-4x = -40

-4x / -4 = -40 / -4

x =10

y = -10 + 3

y = -7

Review

  - 5x + 3y = 1

- 4x + 3y = 5

Addition, subtraction or substitution? Why?

Subtraction, because both equations have the term +3y.

Review

  4x = - 2y + 56 x = - 5y + 59

Addition, subtraction or substitution? Why?

Substitution, because the equations have no terms with the same number.

Review

- x + y = 12x - 3y = - 30

Addition, subtraction or substitution? Why?

Addition, because the first equation has -x and the second has x.

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