System of EquationsA set of two or more equations with the same variables.

To solve a system of equations means to find values for the variables in the equations which make all equations true at the same time.

Example 1Solve by graphing:

••

•••

-2,0)

Types of Linear SystemsConsistent: A system of equations that has at least one solution

Independent: A system of equations that has exactly one solution

Dependent: A system of equations that has infinitely many solutions

Inconsistent: A system of equations that has no solution

Graphs resulting in the same line are said to “coincide”

Which type of systems coincide?

Elimination Method

Substitution MethodA method for solving systems of equations in which one equation is solved for a variable and the substituted into the other equation.

A method for solving systems of equations in which the equations are added vertically, only after ensuring that one of the variables will cancel out.

Example 2

50105

100105.7

yx

yx

50 12.5

_______________

x4x

5( )2( )

7

142

202)4(5.1

2025.1

y

y

y

yx

Solution: (4, 7)

Example 3

2yx

54

45

845

8342

83)2(2

y

y

y

yy

yy

Solve one equation for one variable:

Substitute into the other equation & solve:

Substitute into either equation & solve:

54

54

2

2

2

x

x

yx

Example 4a

200326 mC

1600226 mC

1. Define the variables:

Let m = monthly payment

C = total cost

2. Write the equations (2 variables – 2 equations)

3. Select a method to solve (Graphing, substitution, elimination) & solve.

substitution

14

1400100

1600226200326

m

m

mm4. What does m = 14 mean?

At 14 months, the cost of the lease would be the same either way.

Example 4b

Graph the equations:

Interpret the graph:

Which lease results in the least total cost at 24 months?

The green line has the lowest costs after the first 14 months, which is Lease 2.

HW: Page 70