solving systems of equations
TRANSCRIPT
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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