Chapter 3: Systems of Linear Equations and Inequalities

3.1 Solving Linear Systems by Graphing

Vocabulary

• System of two linear equations

• Solution– an ordered pair (x,y) that satisfies each equation

Ax By C Dx Ey F

Check if the point is a solution( , )0 1

3 2 2x y

x y 2 6

Check if the point is a solution

3 2 2x y x y 2 6

( , )2 2

Solving Systems Graphically2 3 1

3

x y

x y

Check Solutions3 2 6

6 4 12

x y

x y

Check Solutions3 2 6

3 2 2

x y

x y

Number of Solutions of Linear System

Graphical Algebraic

Intersect once One solution

Pair make a single line Infinite solutions

Parallel lines No solution

Don’t intersect

• You plan to work 200 hours this summer mowing lawns and babysitting. You need to make a total of $1300. Babysitting pays $6 per hour and lawn mowing pays $8 per hour. How many hours should you work at each job?

3.2 Solving Linear Systems Algebraically

Substitution Method

1. Solve one equation for one of the variables.

2. Substitute the expression from step 1 into the other equation and solve for the other variable.

3. Substitute the value from step 2 into the revised equation from step 1 and solve.

Solve Using Substitution

3 4 4

2 2

x y

x y

Solve Using Substitution

x y

x y

3 2

4 5 8

Linear Combination Method

1. Multiply one or both equations by a constant to obtain coefficients that are the same except for the sign.

2. Add the revised equations from step 1. Combine like terms to eliminate one of the variables. Solve for the remaining variable.

3. Substitute the value obtained in step 2 into either of the original equations and solve for the other variable.

Solve Using Linear Combination

2 4 13

4 5 8

x y

x y

Solve Using Linear Combination

7 12 22

5 8 14

x y

x y

Solve the Linear System

x y

x y

2 3

2 4 7

Solve the Linear System

6 10 12

15 25 30

x y

x y

• A caterer is planning a party for 64 people. The customer has $150 to spend. A $39 pan of pasta feeds 14 people and a $12 sandwich tray feeds 6 people. How many pans of pasta and how many sandwich trays should the caterer make?

Solve Using Substitution

3 2 10

2 9

x y

x y

Solve Using Substitution

3 7

5 2 12

x y

x y

Solve Using Linear Combination

3 2 6

5 2 18

x y

x y

Solve Using Linear Combination

5 2 12

9 8 19

x y

x y

Solve Using Linear Combination

4 3 0

10 7 2

x y

x y

3.3 Graphing and SolvingSystems of Linear Inequalities

Vocabulary

• System of linear inequalities

• Solution – ordered pair (x,y) that is a solution of each inequality in the system

• Graph – graph of all solutions of the system

x y

x y

6

2 4

Graphing Systems of Inequalities1. Graph the line that corresponds to the

inequality

Dashed line for: < or >

Solid line for:

2. Lightly shade the half-plane that is the graph of each inequality

3. The graph of the system is the region common to all of the half-planes.

or

Graph the systemy x

y x

3 1

2

Graph the systemx

y

y x

0

0

2

Graph the systemx

y x

y x

0

2 1

2 3