Transcript Document

We can apply the fact that the sum of opposites is zero to
systems of equations so that one of the variables can be
eliminated.
Look at the following system of equations. What do you
observe?
x y6
x y2
Zero pair
In each equation, there is a “y”. However, the
signs in the two equations are opposite. If
added together, the “y” and the “-y” would
make a zero pair.
Add the following two
equations together by
adding like terms
together.
x y6
x y2
2x  0  8
Now solve for “x”.
2x = 8
x=4
Now pick one of the two
equations and substitute the
value of “x” into that
equation and solve for “y”.
x+y=6
4+y=6
-4
-4
y=2
Solution:
(4,2)
x y6
x y2
Solution:
(4,2)
To check your work, substitute the values for the variables into
each equation and determine if it is true.
x y6
4 2 6
66
x y2
4 2 2

2 2
The solution is correct.

Solve the system
4x + 3y = 16
2x – 3y = 8
6x
= 24
6
6
x = 4
4x + 3y = 16
2x – 3y = 8
4x + 3y = 16
4(4) + 3y = 16
16 + 3y = 16
-16
-16
3y = 0
3
3
y=0
Solution is (4, 0)
Solve the system 2x + 8y = -30
-2x – 10y = 34
2x + 8y = -30
-2x – 10y = 34
-2y = 4
-2
-2
y = -2
2x + 8y = -30
2x + 8(-2) = -30
2x + -16 = -30
+ 16
+16
2x
= -14
2
2
x = -7
Solution is (-7, -2)
Addition worked when the signs of one of the variables were
opposite. However, you may encounter a system of equations
in which the signs of the variables are the same. In this case,
instead of adding, you will be subtracting one equation from the
other. Remember that subtraction is really the same as adding
the opposite.
4x  y  9
3x  y  6
The signs of the variables are all positive.
Therefore, in order to solve this system, we
can subtract the bottom equation from the
top equation.
Subtract the bottom
equation from the top
equation.
(-)
4x  y  9
3x  y  6
x03
or
Since we have the value for
“x”. Pick one of the equations
and substitute the value 3 for
“x” and solve for “y”.
3x + y = 6
(3)(3) + y = 6
9+y=6
-9
x3
Solution:
-9
y = -3
(3,-3)
4x  y  9
3x  y  6
Solution:
(3,-3)
To check your work, substitute the values for the variables into
each equation and determine if it is true.
4x  y  9
4 ( 3)  ( 3)  9
12  3  9
9  9
3x  y  6
3( 3)  ( 3)  6
9 3 6
66 
The solution is correct.
Solve the system 3x + 5y = 18
3x + 2y = 9
3x + 5y = 18
-( 3x + 2y = 9)
3y = 9
3
3
y= 3
3x + 5 y = 18
3x + 5(3) = 18
3x + 15 = 18
-15
-15
3x
= 3
3
3
x = 1
The solution is (1, 3)
Summary of Steps
Step 1: The equations must
be Standard Form.
Standard Form: Ax + By = C
Step 2: Determine which
variable to eliminate.
Look for variables that have the
same or opposite coefficients.
Step 3: Add or subtract the
equations.
Solve for the Variable.
Step 4: Plug back in to find
the other variable.
Find the value of the second
variable.
Step 5: Check your
solution.
Substitute your ordered pair into
BOTH equations.
Solve each system of equations by using addition or subtraction.
1.
2x  4 y  18
x  4y  3
2.
3 p  2r  5
3 p  6r  15
3. The sum of two numbers is 85. The difference between
the two numbers is 19. What are the two numbers?
2x  4 y  18
x  4y  3
Add the two equations
together.
2x  4 y  18
(+) x  4 y  3
3x  0  21
Solve for “x”.
3x = 21
3 3
x=7
Solve for “y” by substituting
the value for “x” into one of the
equations.
x  4y  3
7  4y  3
7
7
 4 y  4
y1
Solution: (7,1)
3 p  2r  5
3 p  6 r  15
Subtract the bottom
equation from the top
equation.
(−)
3 p  2r  5
3 p  6r  15
 4 r  20
Solve for “s” by substituting the
value for “r” into one of the
equations.
3 p  2r  5
3 p  2(5)  5
3 p  10  5
 10  10
3 p  15
Solve for “r”.
3 p 15

3
3
4 r
20

4
4
r  5
p5
Solution:
(5,-5)
The sum of two numbers is 85. The difference between the
two numbers is 19. What are the two numbers?
x  y  85
x  y  19
2x  0  104
2x 104

2
2
x  52
Write an equation for the sum of the numbers.
Write an equation for the difference of the numbers.
Add the equations together.
Solve for x.
Next substitute x = 52 into one of the equations.
x  y  85
x  y  19
52  y  85
y  33
The two numbers
are 52 and 33.
Solve each system of equations by using addition or subtraction.
(4) x + 2y = 7
3x – 2y = -3
(5) 2x + 7y = 1
2x + 3y = 9
(6) Twice a number minus a second number is
15. The sum of the two numbers is -6.
What are the two numbers?