Download Solving Linear Systems: Addition Method

Solving Linear Systems of Equations - Addition Method
• Recall that to solve the
a1x  b1 y  c1
linear system of equations
a
x

b
y

c
2
2
2
in two variables ...
we need to find the value of x and y that satisfies both
equations. In this presentation the Addition Method will
be demonstrated.
• Example 1:
Solve the following system
of equations ...
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xy4
xy2
Solving Linear Systems of Equations - Addition Method
• Label the equations as
# 1 and # 2.
#1
#2
xy 4
xy 2
• Equation # 2 states that x - y has the same value as - 2.
Since we can add the same value to both sides of an
equation to produce an equivalent equation, we proceed
as follows.
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Slide 2
Solving Linear Systems of Equations - Addition Method
• Start with equation # 1 ...
• Add x - y (the left hand
side of equation # 2) to the
left hand side ...
xy  4
xy
2
2x  0  6
• Then add - 2 (the right
hand side of equation # 2)
to the right hand side ...
• The result is ...
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Slide 3
Solving Linear Systems of Equations - Addition Method
• Solve this equation for x ...
• Now use either equation
# 1 or equation # 2 to find
the value of y. Using
equation # 1 ...
2x  0  6
x 3
xy4
3 y  4
 y 1
• The solution to the system is (3, 1), or
x=3
y=1
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Slide 4
Solving Linear Systems of Equations - Addition Method
• Since equation # 1 was
used in the last step, check
by substituting the solution
values into equation # 2 ...
xy2
3 1  2
22
• Notice that in this system
the coefficients of the y
variables were the same
except for sign (+ 1 and - 1).
xy 4
xy 2
• This is the form that a system must have right before
the addition step. When the equations are added, one
variable is eliminated, and the result is one equation
with one unknown, which is easily solved for.
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Slide 5
Solving Linear Systems of Equations - Addition Method
• Example 2:
Solve the following system
of equations ...
#1
#2
3x  4 y  8
7 x  2 y  3
• Notice that neither variable meets the condition of
coefficients being the same except for sign. This must be
accomplished before proceeding.
• Multiplying equation # 2
by + 2 yields ...
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3x  4 y  8
14x  4 y  6
Slide 6
Solving Linear Systems of Equations - Addition Method
• Now the coefficients on y
are the same, except for
sign ...
• Addition of the equations
yields ...
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3x  4 y  8
14x  4 y  6
17 x  2
2
x 
17
Slide 7
Solving Linear Systems of Equations - Addition Method
• Substitute the value for x
into equation # 1 (either
equation could be used at
this point) ...
and solve for y ...
• The solution to the system is ...
2
x
17
65
y
34
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2
x
17
3x  4 y  8
2
3   4 y  8
 17 
6  68y  136
65
y
34
Slide 8
Solving Linear Systems of Equations - Addition Method
• Example 3:
Solve the following system
of equations ...
#1
#2
3x  12 y  9
7 x  5y  2
• Neither variable meets the condition of coefficients being
the same except for sign. The equations can be multiplied
by constants to achieve this goal for either variable.
• The coefficients of x are 3 and 7. The lcm of 3 and 7
is 21, so multiply each equation by a convenient value to
get coefficients of 21, opposite in sign.
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Slide 9
Solving Linear Systems of Equations - Addition Method
• Multiply equation # 1 by 7 ...
and equation # 2 by - 3 ...
• Adding the two equations
yields ...
3x  12 y  9
21x  84 y  63
7 x  5y  2
 21x  15y  6
21x  84 y  63
 21x  15y  6
 99 y  57
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Slide 10
Solving Linear Systems of Equations - Addition Method
19
y
33
• The solution for y is ...
• Substituting this value for
y in equation # 2 (either
equation could be used
here) yields ...
23
x
33
and the solution is ...
23
x
33
19
y
33
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Slide 11
Solving Linear Systems of Equations - Addition Method
• Summary:
1) Multiply the equations by constants so that one
of the variables will have the same coefficients,
opposite in sign.
2) Add the left sides and then the right sides of
the two equations to yield one equation in one
variable, for which we can solve.
3) Substitute the given value for the variable into
either equation and solve for the other variable.
4) To check the solutions, substitute both values
into the equation that was not used in step 3.
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Slide 12
Solving Linear Systems of Equations - Addition Method
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