# Systems of Linear Equations (2007)

Chapter 8

 A relationship between an independent and a independent variable in which as the independent variable changes the dependent variable changes by a constant amount.

Examples:

Profit made based on the number of tickets sold to a dance

Constant population growth over time

Cost for an appliance repair based on a set fee and an hourly price

Two relations that represent a comparison with the same information

A set of equations with the same variables

Two lines in the same coordinate plane

Text p. 454

World Records

Let s = swim time f = float time

Which is independent?

An equation to represent the record holders time: f + 3f = 44

An equation to represent the amount of time available to swim and float: f + s = 24

 S + 3f = 44

 S + f = 24

Graphs

Equations

Mappings

Ordered Pairs

Tables

Lines in a plane can : be Parallel…Never intersect

 These lines will have no solution

Intersect at only one point

 These lines will yield one solution be Co-linear…the same line (one line a scale factor of the other)

 These lines will have an infinite number of solutions

1.) x + 2y = 1

2x + 5 = y

2.) 3x – y = 2

12x – 4y = 8

3.) x – 2y = 4 x = 2y - 2

 Intersect or are co-linear are said to be consistent because there is at least one ordered pair (point) common to both lines.

 Co-linear have an infinite number of common points!

Are parallel are said to be inconsistent because there is not one point common to both lines

If a system has exactly one solution , it is independent, so …. Intersecting lines are independent!!!

If a system has an infinite number of solutions then it is dependent …… Co-linear lines are dependent!!!

These terms DO NOT apply to Parallel Lines

1.) y = 3x – 4 y = -3x + 4

2.) x + 2y = 5

2x + 4y = 2

* Check with graphing calculator

3.) y = -6

4x + y = 2

4.) 2x + 3y = 4

-4x – 6y = -8

The solution when graphing may not be exact

Example: p. 262 Census problem

Substitution

Elimination

 Multiplication and Division

Solve for one of the variables in one of the two equations

Which one??

The one with a coefficient of 1 or with the easiest coefficient to solve for

Substitute the expression equal to the variable into the other equation and solve for the other variable

Use this value to find the value for the original variable.

1.) x + 4y = 1

2x – 3y = -9

2.) 5/2x + y = 4

5x + 2y = 8

3.) 3x + 4y = 7

3/2x + 2y = 11

EJH Labs needs to make

1000 gallons of a 34% acid solution. The only solutions available are

25% acid and 50% acid.

How many gallons of each solution should be mixed to make the

34% solution?

A metal alloy is 25% copper. Another metal alloy is 50% copper.

How much of each alloy should be used to make 1000 grams of metal alloy that is 45% copper?

Subtraction

Use this method when one of the variables’ coefficients in the two equations is the same or are additive inverses

Add or subtract the equations to eliminate a variable

 Examples:

1.) ex p. 469

2a + 4c = 30

2a + 2c = 21.5

2.) 3x – 2y = 4

4x + 2y = 4

The sum of two numbers is 18. The sum of the greater number and twice the smaller number is 25. Find the numbers.

 The sum of two numbers Is 27. Their difference is 5. Find the numbers.

 Lena is preparing to take the SATs. She has been taking practice tests for a year and her scores are steadily improving. She always scores about 150 higher on math than she does on verbal. She needs a 1270 to get into the college she has chosen. If she assumes that she will still have that 150 difference between the two tests, what will she have to score on each part?

Multiplication and

Division (Scaling)

Use this method when all variables have different coefficients

 Example p. 475

75p + 30n = 40.05

50p + 60n = 35.10

1.) 2x + 3y = 5

5x + 4y = 16

2.) 3x + 5y = 11

2x + 3y = 7

3.) 2x – 3y = 8

-5x + 2y = 13

A bank teller reversed the digits in the amount of a check and overpaid the customer by \$9. The sum of the digits in the two digit amount was 9 Find the amount of the check.

Example 2 and 3 p. 477

 Graphing?

 Substitution?

 Elimination?