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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