3.1 Solve Linear Systems by Graphing

3.1 Solve Linear Systems by Graphing

Goal Solve systems of linear equations.

Your Notes

VOCABULARY

System of two linear equations

Two equations, with the variables x and y that can be written as:

Ax + By = CEquation 1

Dx + Ey = FEquation 2

Solution of a system

An ordered pair (x, y) that satisfies each equation

Consistent

A system that has at least one solution

Inconsistent

A system that has no solution

Independent

A consistent system that has exactly one solution

Dependent

A consistent system that has infinitely many solutions

Example 1

Solve a system graphically

Graph the system and estimate the solution. Then check the solution algebraically.

4x + 2y = 4Equation 1

2x 3y = 10Equation 2

Solution

Graph both equations. The lines appear to intersect at (_2_, _2_). Check this algebraically as follows:

Equation 1 / Equation 2
4x + 2y = 4 / 2x 3y = 10
4(_2_) + 2(_2_) = ? =4 / 2(_2_) 3(_2_) =? = 10
_4_ = 4  / _10_ = 10 

Your Notes

CheckpointGraph the linear system and estimate the solution. Then check the solution algebraically.

1. 4x + y = 2

6x 3y = 12

(1, 6)

NUMBER OF SOLUTIONS OF A LINEAR SYSTEM

Exactly one solution / Infinitely many solutions / No Solutions
Lines intersect at one point consistent and independent Infinitely many solutions it y / Lines coincide; consistent and dependent / Lines are _parallel_;
_inconsistent_

Example 2

Solve a system with many solutions

Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent

2x + y = 4Equation 1

4x 2y = 8Equation 2

The graphs of the equations are _the same line_. So, each point on the line is a solution, and the system has _infinitely many_ solutions. Therefore, the system is _consistent and dependent_.

Your Notes

Example 3

Solve a system with no solution

Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent.

2x + 4y = 8Equation 1

2x + 4y = 4Equation 2

Solution

The graphs of the equations are two _parallel lines_. Therefore, the system is _inconsistent_.

Checkpoint Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent

1. 3x 2y = 6

5x + 4y = 8

(4, 3);

consistent and independent

1. x2y = 5

2x4y = 10

infinitely many solutions; consistent and dependent

1. 6x3y = 12

6x3y = 6

no solutions; inconsistent

1. x + y = 2

4x3y = 1

(1, 1); consistent and independent

Your Notes

Example 4

Writing and using a linear system

Ice Cream Shop At an ice cream shop, one customer pays $9 for 2 sundaes and 2 milkshakes. A second customer pays $13 for 2 sundaes and 4 milkshakes. How much do each sundae and milkshake cost?

Verbal model

=+

_9_ = _2_ x+_2_ yEquation 1 (Customer 1)

_13_ = _2_ x + _4_ yEquation 2 (Customer 2)

Graph the equations

_2_ x + _2_ y = _9_ and

_2_ x + _4_ y = _13_.

The lines appear to intersect at about the point (_2.5_, _2_).

Check this algebraically.

_2_(_2.5_) + _2_(_2_) = _5_ + _4_ = 9 Equation 1 checks.

_2_(_2.5_) + _4_(_2_) = _5_ + _8_ = 13  Equation 2 checks.

The solution is (2.5, _2_). So, each sundae costs $ 2.50 and each milkshake costs $ 2.00.

CheckpointComplete the following exercise.

1. In Example 4, how much do each sundae and milkshake cost if the first customer pays $7 and the second customer pays $10?

sundae: $2,

milkshake: $1.50

Homework

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