Systems of Linear Equations or Simultaneous Equations

Chapter 8

What is a linear relation?

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

A System of Linear Relations is

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

An Example of a System of Equations

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

System and the graph

S + 3f = 44

S + f = 24

Linear Relations can be represented as:

Graphs Equations Mappings Ordered Pairs Tables

Graphing Systems of Equations

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

Examples:

1.) x + 2y = 1 2x + 5 = y

2.) 3x – y = 2 12x – 4y = 8

3.) x – 2y = 4 x = 2y - 2

Lines that…

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

Still more Vocab….

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

Still More examples:

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

Problem with graphing….

The solution when graphing may not be exact

Example: p. 262 Census problem

Other methods for solving systems of equations

Substitution Elimination

Addition and Subtraction Multiplication and Division

How to use Substitution

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.

Examples using substitution

1.) x + 4y = 1 2x – 3y = -9

2.) 5/2x + y = 4 5x + 2y = 8

3.) 3x + 4y = 7 3/2x + 2y = 11

But still more examples….(your favorite!)

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?

When and How to use Elimination

Addition and 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

Still more Examples…

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.

And More!!

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?

When and How to use Elimination

Multiplication and Division (Scaling)

Use this method when all variables have different coefficients

Example p. 475

75p + 30n = 40.05 50p + 60n = 35.10

More Examples…

1.) 2x + 3y = 5 5x + 4y = 16

2.) 3x + 5y = 11 2x + 3y = 7

3.) 2x – 3y = 8 -5x + 2y = 13

Still More Examples:

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

So which method should you use??

Graphing?

Substitution?

Elimination?