(c) MathScience Innovation Center 2007 Solving Linear Systems Trial and Error Substitution Linear Combinations (Algebra) Graphing

(c) MathScience Innovation Center 2007

Solving Linear SystemsTrial and ErrorSubstitutionLinear Combinations (Algebra)Graphing


Linear System

• Two or more equations

• Each is a straight line

• The solution = points shared by all equations of the system


Linear System

• There may be one solution

• There may be no solution

• There may be infinite solutions


Linear System

• Consistent= there is a solution

• Inconsistent= there is no solution

• Independent= separate, distinct lines

• Dependent= same line


Linear System

• Consistent, independent

• Inconsistent, independent

• Consistent, dependent


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)


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.


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


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


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


Substitution

•So x = 1.

•Substitute for y in bottom equation:

• y = 2 – 3x

• y = 2 – 3(1)

•Y = -1

•Final solution: ( 1, -1)


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?


Linear Combinations(Algebra)

Example:

6x – 4y = 10

3x + y = 2Multiply this equation by -2 or 4



Example:

6x – 4y = 10

3x + y = 2 Multiply this equation by -2 or 4

Multiplying by -2 yields

6x – 4y = 10

-6x + -2y = -4+


Yes, x- 6 y = 6



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



Multiplying by 4:

6x – 4y = 10

12x + 4y = 8+


Yes, y18 x = 18

Now, x = 1. Substitute x = 1 to find y.

6 (1) – 4y = 10

12 (1) + 4y = 8So, y = -1



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


Graphing

Graph each equation:

6x – 4y = 10

3x + y = 2Note: this problem is difficult because the equations are not solved for y


Graphing


6x – 4y = 10

3x + y = 2So it might be easiest to hand plot using the x and y intercepts.


Graphing


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.


Graphing

Y1 = (10-6x)/(-4)

Y2 = 2- 3x


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.



y = 2 x - 4

y = ¾ x + 5

A. Substitution

B. Linear

Combinations

(algebra)

C. Graphing

Why?

Both equations are already solved for y.



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.



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.


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


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

Documents

(c) MathScience Innovation Center 2007 Solving Linear Systems Trial and Error Substitution Linear Combinations (Algebra) Graphing