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Chapter 8 Section 3
Solving System of Equations
by the Addition Method
Learning Objective
Solve a system of equations by the Addition
Method
Key Vocabulary:
Addition (or elimination) method
Solve Systems of Equations
Addition Method
1.
If necessary rewrite each equation so that the terms containing variables
appear on the left side of the equal sign and the constants appear on the
right side of the equal sign
2.
If necessary multiply one or both equations by a constant(s) so that when
the remaining equations are added the resulting sum will contain only
one variable
3.
Add the equations resulting in one equation
4.
Solve for the variable
5.
Substitute the value found into one of the original equations, and solve
that equation for the other variable
6.
Check
Solve Systems of Equations
Addition Method
Example:
Step 1:
Variables are already
on one side and
constants are on the
other side
Equation 1
x+y=7
Step 4
Solve for the variable
5x = 10
x = 10/5
x=2
Equation 2
4x – y = 3
Step 5
Substitute the value
found into one of the
original equations, and
solve that equation for
the other variable
x+y=7
2+y=7
y=7–2
y=5
Step 2
adding will already result
in one variable
Step 3
Add the equations
resulting in one equation
4x – y = 3
x+y= 7
5x
= 10
Solution is (2, 5)
Consistent
What do you think the ordered pair (2,5) represents on the graph?
Solve Systems of Equations
Addition Method
Example:
Step 1:
Variables are already
on one side and
constants are on the
other side
Equation 1
x + 2y = 2
Step 4
Already solved for the
variable
Equation 2
x + 3y = 6
Step 2
multiply one or both
equations by a constant(s)
-1(x +2y = 2)
-x – 2y = -2
Step 5
Substitute the value
found into one of the
original equations, and
solve that equation for
the other variable
Step 3
Add the equations
resulting in one equation
x + 3y = 6
-x – 2y = -2
y=4
x + 2y = 2
x + 2(4) = 2
x+8=2
x=2–8
x = -6
Solution is (-6, 4)
Consistent
Both sides of the equation has to be multiplied by the given number.
Solve Systems of Equations
Addition Method
Example:
Step 1:
Variables are already
on one side and
constants are on the
other side
Equation 1
3x + y = 8
Step 2
multiply one or both
equations by a constant(s)
-3(x + 5y = -2)
-3x – 15y = 6
Step 3
Add the equations
resulting in one equation
3x + y = 8
-3x – 15y = 6
-14y = 14
Equation 2
x + 5y = -2
Step 4
Solve for the variable
-14y = 14
y = -1
Step 5
Substitute the value
found into one of the
original equations, and
solve that equation for
the other variable
x + 5y = -2
x + 5(-1) = -2
x - 5 = -2
x = -2 + 5
x=3
Solution is (3, -1)
Consistent
Both sides of the equation has to be multiplied by the given number.
I could have multiplied the first equation by -5 and got the same results.
Solve Systems of Equations
Addition Method
Example:
Step 1:
Variables are already
on one side and
constants are on the
other side
Step 2
multiply 1st equation by 8
and the 2nd equation by -2
Equation 1
2x + 9y = 5
Equation 2
8x - 3y = -6
8(2x + 9y = 5)
16x + 72y = 40
-2(8x – 3y = -6)
-16x + 6y = 12
Step 3
Add the equations
resulting in one equation
16x + 72y = 40
-16x + 6y = 12
78y = 52
Step 4
Solve for the variable
78y = 52
y = 52/78
y=⅔
Step 5
Substitute the value
found into one of the
original equations, and
solve that equation for
the other variable
8x - 3y = -6
8x – 3(2/3) = -6
8x - 2 = -6
8x = -6 + 2
8x = -4
x = -4/8
x=-½
Solution is (- ½ , ⅔)
Consistent
(26)
Solve Systems of Equations
Addition Method
Example:
Step 1:
Variables are already
on one side and
constants are on the
other side
Equation 1
3x + 5y = -6
Step 2
multiply 1st equation by 2
and the 2nd equation by 3
2(3x + 5y = -6)
6x + 10y = -12
Equation 2
-2x + 7y = 4
Step 4
Solve for the variable
31y = 0
y = 0/31
y=0
Step 5
Substitute the value
found into one of the
original equations, and
solve that equation for
the other variable
3x + 5y = -6
3x – 5(0) = -6
3x = -6
3x = -6/3
x = -2
3(-2x + 7y = 4)
-6x + 21y = 12
Step 3
Add the equations
resulting in one equation
6x + 10y = -12
-6x + 21y = 12
31y = 0
Solution is (- 2 , 0)
Consistent
Solve Systems of Equations
Addition Method
Example:
Step 1:
Variables are already
on one side and
constants are on the
other side
Step 2
multiply 1st equation by 2
and the 2nd equation by 3
Step 3
Add the equations
resulting in one equation
Equation 1
3x - 2y = 1
Equation 2
-6x + 4y = 5
2(3x - 2y = 1)
6x - 4y = 2
6x + 4y = 5
-6x – 4y = 2
0=7
False, No solution
Inconsistent
Parallel lines
Slope intercept form would
show us that they have the
Same slope and
different y-intercepts
-2y = -3x + 1
y = 3/2 x – ½
4y = 6x + 5
y = 3/2 x + 5/4
Solve Systems of Equations
Addition Method
Example:
Step 1:
Variables on one side
and constants are on
the other side
Equation 1
y = 1/3 x + 2
-1/3 x + y = 2
Equation 2
3y – x = 6
-x + 3y = 6
Step 2
multiply 1st equation by 2
and the 2nd equation by 3
-3(-1/3x + y = 2)
x – 3y = -6
Step 3
Add the equations
resulting in one equation
x – 3y = -6
-x + 3y = 6
0=0
The answer would have
been clear if we had put
the 2nd equations in slope
intercept form. We could
then see that they have the
same slope and same
y-intercept
3y = x + 6
y = 1/3 x + 2
True, same line
Dependent
Infinite number of solution
Solve Systems of Equations
Addition Method
Example:
Step 1:
Variables are already
on one side and
constants are on the
other side
Equation 1
4x + 5y = 3
Step 2
multiply 1st equation by 2
and the 2nd equation by 3
-2(2x - 3y = 1)
-4x + 6y = -2
Step 3
Add the equations
resulting in one equation
Equation 2
2x - 3y = 1
4x + 5y = 3
-4x + 6y = -2
11y = 1
Step 4
Solve for the variable
Step 5
Substitute the value
found into one of the
original equations, and
solve that equation for
the other variable
11y =1
y = 1/11
2x - 3y = 1
2x - 3(1/11) = 1
2x – 3/11 = 1
2x = 3/11 + 1
2x = 14/11
x = 14/11 (½)
x = 14/22
x = 7/11
Solution is (7/11 , 1/11)
Consistent
Remember

The objective is to obtain two equations who
sum will be an equation containing only one variable

Contemplate which variable will be easiest to eliminate and what
multiplication will be needed to make the elimination possible

Neatness and organization will help when solving by substitution and by
addition method

There is always more than one way to solve a problem

The solution should be and ordered pair.

Check by substituting the solution back into the original equations.
HOMEWORK 8.3
Page 511 - 512
#7, 9, 11, 13, 17, 19, 23, 33