Download Solving Linear Systems

Solving Linear
Systems
Trial and Error
Substitution
Linear Combinations (Algebra)
Graphing
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Linear System
• Two or more equations
• Each is a straight line
• The solution = points shared by all
equations of the system
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Linear System
• There may be one solution
• There may be no solution
• There may be infinite solutions
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Linear System
• Consistent= there is a solution
• Inconsistent= there is no solution
• Independent= separate, distinct
lines
• Dependent= same line
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Linear System
• Consistent, independent
• Inconsistent, independent
• Consistent, dependent
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Trial and Error
• Try any point and see if it satisfies
every equation in the system
(makes each equation true)
Example:
6x – y = 5
3x + y = 13
Try ( 2,7) and try ( 1,10)
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Trial and Error Conclusion:
Since
(2,7)
Try ( 2,7)
works and
+
6 (2) – (7) = 5
(1,10) does
3 (2) + 7 = 13 + not work,
(2,7) is a
solution to
Try ( 1,10)
6 (1) – 10 = 5 X the system
and (1,10) is
3 (1) + 10 = 13 +
not a
solution.
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Substitution
• Solve one equation for one variable
and substitute into the other
equations.
• Hint: Easiest to solve for a variable
with a coefficient of 1
Example:
6x – 4y = 10
3x + y = 2
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Substitution
Example:
6x – 4y = 10
3x + y = 2
Solve for y in bottom equation:
6x – 4y = 10
y = 2 – 3x
Substitute for y in top equation:
6x – 4(2-3x) = 10
y = 2 – 3x
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Substitution
Substitute for y in top equation:
6x – 4(2-3x) = 10
y = 2 – 3x
Simplify top equation and solve
for x:
•6x – 4(2-3x) = 10
•6x – 8 + 12 x = 10
•18 x = 18
•18x/18 = 18/18
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Substitution
•So x = 1.
•Substitute for y in bottom
equation:
• y = 2 – 3x
• y = 2 – 3(1)
•Y = -1
•Final solution: ( 1, -1)
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Substitution
•Check your work:
•Final solution: ( 1, -1)
Example:
6x – 4y = 10
3x + y = 2
Example:
6(1) – 4( -1) = 10 +
3(1) + -1 = 2 +
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Linear Combinations
(Algebra)
• Try adding the equations together
so that at least one variable
disappears
• Hint: You can multiply any
equation by an integer to insure
this happens ! If we draw a bar
Example:
and add
does any variable
6x – 4y = 10
disappear?
+ 3x + y = 2
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Linear Combinations
(Algebra)
Example:
6x – 4y = 10
3x + y = 2
Multiply this
equation by -2
or 4
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Linear Combinations
(Algebra)
Example:
6x – 4y = 10
3x + y = 2
Multiply this equation
by -2 or 4
If we draw a bar
Multiplying by -2 yields
and add
does any variable
6x – 4y = 10
disappear?
-6x + -2y = -4
+
-6y=6
Yes, x
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Linear Combinations
(Algebra)
Example:
6x – 4y = 10
3x + y = 2
Since - 6 y = 6,
y = -1
Now, use substitution to find x
6x – 4 (-1) = 10
3x + (-1) = 2
X=1
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Linear Combinations
(Algebra)
Multiplying by 4:
6x – 4y = 10
12x + 4y = 8
If we draw a bar
and add
does any variable
disappear?
+
18 x
= 18
Yes, y
Now, x = 1. Substitute x = 1 to find y.
6 (1) – 4y = 10
12 (1) + 4y = 8
So, y = -1
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Linear Combinations
(Algebra)
One last question
6x – 4y = 10
3x + y = 2
Is it easier to
multiply this
equation by -2
or 4 ?
Most people are more successful
when using positive numbers 
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Graphing
Graph each equation:
6x – 4y = 10
3x + y = 2
Note: this
problem is
difficult
because the
equations are
not solved for y
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Graphing
Graph each equation:
6x – 4y = 10
3x + y = 2
So it might be
easiest to hand
plot using the x
and y intercepts.
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Graphing
Graph each equation:
6x – 4y = 10
3x + y = 2
To use a
graphing
Y1 = (10-6x)/(-4) calculator,
solve for y.
Y2 = 2- 3x
Simplifying is not necessary.
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Graphing
Y1 = (10-6x)/(-4)
Y2 = 2- 3x
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Which is the easiest method
to solve this system?
x=4
A. Substitution
2x + 3 y = 14 B. Linear
Combinations
Why?
(algebra)
One equation
C.
Graphing
is already
solved for x,
ready for
substitution.
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Which is the easiest method
to solve this system?
y=2x-4
y= ¾x+5
Why?
Both
equations are
already
solved for y.
A. Substitution
B. Linear
Combinations
(algebra)
C. Graphing
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Which is the easiest method
to solve this system?
3 x – 2 y = 14 A. Substitution
4x + 2 y = 21 B. Linear
Combinations
Why?
(algebra)
When you add
C.
Graphing
them
together, the y
disappears.
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Which is the easiest method
to solve this system?
x – 9 y = 10 A. Substitution
2x + 3 y = 7 B. Linear
Combinations
Why?
(algebra)
Substitution
C.
Graphing
would not be
difficult either,
but graphing
would be more
difficult.
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If you use linear
combinations, what would
you multiply by and which
equation would you use?
x – 9 y = 10
2x + 3 y = 7
Which
might be a
wee tiny bit
easier?
A. Top equation
by -2
B. Bottom
equation by 3
B. Working with positive numbers
may lead to fewer errors
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Match a system to the
easiest solution method.
A
B
Y = 2x + 1
Y = 1/3 x - 9
y = 2x + 1
4x – 19 y = 34
Substitution
Linear Combinations
(Algebra)
Graphing
C
3 x – 5 y = 26
- 3 x + 4 y = 17
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