8.3 The Addition Method
Also referred to as Elimination Method
The Addition Method
1) Write the second equation below the first.
2) Add the equations together and solve for
the remaining variable.
x-y=1
x-y=1
x-y=1
x -xy- =
1xy1-=xy1-=y1= 1
xy -=
Solve: x + y = 5
+
2x
=6
x=3
3+y=5
y=2
(3,2)
x+y=5
2x - y = 4
3x
=9
x=3
x+y=5
3+y=5
y=2
(3,2)
2 Special Cases
Case 1: NO SOLUTION
• Both variables cancel out but the constant
does not
• Leaving a false equation.
4x - 2y = 2
No
solution
-4x + 2y = -16
0 + 0 = -14
Case 2: INFINITELY MANY
SOLUTIONS
• Both variables and the
constant cancel out
• Leaving a true
equation
5x - 7y = 6
-5x + 7y = -6
0+0=0
Infinitely
many
solutions
They all do not cancel out so easily
If the equations do not eliminate a variable
when you add them together:
• Multiply the whole equation by a number
that will help you cancel it out.
4x - 2y = 7
2 3x + y = 4
4x - 2y = 7
6x + 2y = 8
10x
3(1.5) + y = 4
4.5 + y = 4
y = - 0.5
(1.5,-0.5)
= 15
x = 1.5
2x + 3y = 8
-1 x + 3y = 7
2x + 3y = 8
-x - 3y = -7
x
2(1) + 3y = 8
3y = 6
y=2
(1,2)
=1
3 4x + 2y = 18
4 -3x + 5y = 6
12x + 6y = 54
-12x + 20y = 24
26y = 78
-3(3) + 5y = 6
y=3
-9 + 5y = 6
5y = 15
y=3
(3,3)
3 5x + 3y = 2
-5 3x + 7y = -4
15x + 9y = 6
-15x - 35y = 20
-26y = 26
3x + 7(-1) = -4
y = -1
3x- 7 = -4
3x= 3
x=1
(1,-1)
5(a-b)=10 and a+b=2
The sum of two numbers is 72. The
difference is 58. Find the numbers.
x + y = 72
x - y = 58
7 and 65
The sum of the length and width of a rectangle
is 25 cm. The length is 2 less than twice the
width. Find the length and width.
 2w  2L  w  25
3w 2  25
3w  27
w9
L  2w  2
L  2 9  2
L  16
Assignment:
Page 371
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