solving linear systems
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(c) MathScience Innovation Center 2007
Solving Linear SystemsTrial and ErrorSubstitutionLinear 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
Try ( 2,7)
6 (2) – (7) = 5
3 (2) + 7 = 13
Try ( 1,10) 6 (1) – 10 = 5
3 (1) + 10 = 13
++
+X
Conclusion:
Since (2,7) works and (1,10) does not work, (2,7) is a solution to the system and (1,10) is 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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SubstitutionExample:
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
Simplify top equation and solve for x:
•6x – 4(2-3x) = 10
•6x – 8 + 12 x = 10
•18 x = 18
•18x/18 = 18/18
Substitute for y in top equation:
6x – 4(2-3x) = 10
y = 2 – 3x
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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 !
Example:
6x – 4y = 10
3x + y = 2+
If we draw a bar and add does any variable disappear?
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Linear Combinations(Algebra)
Example:
6x – 4y = 10
3x + y = 2Multiply 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
Multiplying by -2 yields
6x – 4y = 10
-6x + -2y = -4+
If we draw a bar and add does any variable disappear?
Yes, x- 6 y = 6
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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 x6x – 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?
Yes, y18 x = 18
Now, x = 1. Substitute x = 1 to find y.
6 (1) – 4y = 10
12 (1) + 4y = 8So, y = -1
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Linear Combinations(Algebra)
One last question
6x – 4y = 10
3x + y = 2Is 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 = 2Note: 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 = 2So 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 calculator, solve for y.
Y1 = (10-6x)/(-4)
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
2x + 3 y = 14
A. Substitution
B. Linear
Combinations
(algebra)
C. Graphing
Why?
One equation is already solved for x, ready for substitution.
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Which is the easiest method to solve this system?
y = 2 x - 4
y = ¾ x + 5
A. Substitution
B. Linear
Combinations
(algebra)
C. Graphing
Why?
Both equations are already solved for y.
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Which is the easiest method to solve this system?
3 x – 2 y = 14
4x + 2 y = 21
A. Substitution
B. Linear
Combinations
(algebra)
C. Graphing
Why?
When you add them together, the y disappears.
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Which is the easiest method to solve this system?
x – 9 y = 10
2x + 3 y = 7
A. Substitution
B. Linear
Combinations
(algebra)
C. Graphing
Why?
Substitution 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 = 7A. Top equation
by -2
B. Bottom equation by 3
Which might be a wee tiny bit easier?
B. Working with positive numbers may lead to fewer errors
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Match a system to the easiest solution method.
Y = 2x + 1
Y = 1/3 x - 9
Substitution
Linear Combinations
(Algebra)
Graphing
A
B
C
y = 2x + 1
4x – 19 y = 34
3 x – 5 y = 26
- 3 x + 4 y = 17
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