Chapter 3 – Systems of Linear Equations

Using a Graph to Solve Linear Systems of Equations

One Day

A system of linear equations is simply two or more linear equations using the same variables.

If the system of linear equations has a solution, then the solution will be an ordered pair (x, y) where x and y make both equations true at the same time.

We will be dealing with systems of 2 equations with 2 variables as well as systems that have 3 equations and 3 variables (x, y, z).

What is a System of Linear Equations?

Cases for System of Equations

Consider the following system:

Solving Systems by Graphing

52

1

yx

yx

25

2

1xy

xy

x

y

(1 , 2)

Anytime we solve a system of equations we must check our solution.

We will do this by substituting the solution back into each equation for x and y. x – y = –1

(1) – (2) = –1

x + 2y = 5

(1) + 2(2) =

1 + 4 = 5

Graphing to Solve a Linear System

512 2

23 1

y x

y x

=- +

= -

Solve the following system by graphing: 3x + 6y = 15

–2x + 3y = –3

Using the slope intercept form of these equations, we can graph them carefully on graph paper.

x

y

Label the

solution!

(3 , 1)

Lastly, we need to verify our solution is correct, by substituting (3 , 1).

( ) ( )3 3 6 1 15+ = ( ) ( )2 3 3 1 3- + =-

1. Put each equation into y=mx+b (solve for y)

2. Graph each equation on graph paper. Precision is important, use a ruler!!

3. Determine the point of intersection, estimate if necessary.

4. Check your solution in both equations!

Steps for Solving by Graphing

Solve the following system of equations by graphing.

Your turn…

22

2

yx

yx

Solve the following system of equations by graphing.

Your turn…

1623

42

yx

yx

pg 120 (# 1-9 odd, 25-35 odd, 44)

Homework

Solving by Substitution and Elimination

Four Days

Solving Linear Systems

There are two methods of solving a system of equations algebraically:

◦ Elimination◦ Substitution

Steps for Solving by Substitution

To solve a system of equations by substitution…

1. Solve one equation for one of the variables.

2. Substitute the value of the variable into the other equation.

3. Simplify and solve the equation for the remaining variable.

4. Substitute back into either equation to find the value of the other variable.

Substitution●Solve the system: x - 2y = -5

y = x + 2 Notice: One equation is already solved for one variable. Substitute (x + 2) for y in the first equation.

x - 2y = -5 x - 2(x + 2) = -5

●We now have one equation with one variable. Simplify and solve. x - 2x – 4 = -5 -x - 4 = -5

-x = -1 x = 1

●Substitute 1 for x in either equation to find y.y = x + 2 y = 1 + 2 so y = 3

●The solution is (1, 3)

Substitution

●Let’s check the solution. The answer (1, 3) must check in both equations.

x - 2y = -5 y = x + 21 - 2(3) = -5 3 = 1 + 2

-5 = -5 3 = 3

Substitution

Solve the systems by substitution:

1. x = 4

2x - 3y = -19

2. 3x + y = 7

4x + 2y = 16

3. 2x + y = 5

3x – 3y = 3

4. 2x + 2y = 4

x – 2y = 0

1. Write each equation in standard form Ax+By=C. 2. Determine which variable you want to eliminate. 3. Multiply an entire equation by a value that will

result in the terms you want to eliminate being additive inverses.

4. Add the equations. The result is one equation with one variable.

5. Solve the resulting equation. 6. Substitute the solution into one of the original

equations and solve for the remaining variable.

Steps for Solving by Elimination

Elimination

● Solve the system: 3s - 2t = 10 4s + t = 6

We could multiply the second equation by 2 and the t terms would be inverses. OR

We could multiply the first equation by 4 and the second equation by -3 to make the s terms inverses.

● Let’s multiply the second equation by 2 to eliminate t. (It’s easier.)3s - 2t = 10 3s – 2t = 10

2(4s + t = 6) 8s + 2t = 12● Add and solve: 11s + 0t = 22

11s = 22 s = 2

● Insert the value of s to find the value of t 3(2) - 2t = 10 t = -2● The solution is (2, -2).

EliminationSolve the system by elimination:1. -4x + y = -12

4x + 2y = 6

2. 5x + 2y = 12-6x -2y = -14

3. 5x + 4y = 12 7x - 6y = 40

4. 5m + 2n = -8 4m +3n = 2

pg 128 (# 1-7 odd, 19-27 odd, 31-37 odd)

Homework

The sum of two numbers is 70 and their difference is 24. Find the two numbers.

Applications of Systems of Equations

Two angles are supplementary. The measure of one angle is 10 degrees more than three times the other. Find the measure of each angle.

Applications of Systems of Equations

Two groups go on a whitewater rafting trip. Group 1 rented 6 rafts and 8 kayaks for a total of $510. Group 2 rented 3 rafts and 11 kayaks for a total of $465. How much did it cost to rent each raft and each kayak?

Applications of Systems of Equations

Applications #1

Homework

Applications #2

Homework

Applications #3

Homework

Systems of Linear Inequalities

Two Days

Complete the following warm-up while I check homework:

Graph the following inequalities:

Warm-up

63 yx

231 xy

Solving Systems of Linear Inequalities

1. We show the solution to a system of linear inequalities by graphing them and determining the region that satisfies all of the individual inequalities simultaneously.

a. This process is easier if we put the inequalities into Slope-Intercept Form, y = mx + b

2. Graph the line using the y-intercept & slope.a. If the inequality is < or >, make the lines dotted.b. If the inequality is < or >, make the lines solid.

3. The solution also includes points not on the line, so you need to shade the region of the graph:

a. above the line for ‘y >’ or ‘y ’.b. below the line for ‘y <’ or ‘y ≤’.

Solve and graph the following system of inequalities.

Example

22

443

yx

yx

1

1

21

43

xy

xy

Solving Systems of Linear Inequalities

a: 3x + 4y > - 4

3: 1

4 a y x

Solving Systems of Linear Inequalities

a: 3x + 4y > - 4

b: x + 2y < 2

3: 1

4 a y x

1: 1

2 b y x

Solving Systems of Linear Inequalities

a: 3x + 4y > - 4 b: x + 2y < 2

The region that satisfies both equations is the area of overlap. This is the solution to our system of inequalities. Any point in this region satisfies the system.

Lets solve the following:

Solving Systems of Linear Inequalities

1

3

yx

xy

Jane’s band wants to spend no more than $575 recording their CD. The studio charges at least $35 per hour to record. Write and graph a system of inequalities to represent this situation.

Applications

Jane’s Band…

xy

y

35

575

The most Dave can spend on hot dogs and bun for a cookout is $42. A package of 10 hot dogs costs $3.50. A package of 8 buns costs $2.50. He needs to buy at least 40 hot dogs and 40 buns. ◦ Write and graph a system of inequalities that

describes this situation.

◦ Give 3 examples of different purchases he can make and still satisfy the requirements.

Applications

Dave’s Cookout

5

4

124.1

y

x

xy

Graph the following system of linear inequalities:

Your Turn

2

43

xy

xy

Graph the following system of linear inequalities:

Your Turn

2x

xy

Graph the following system of linear inequalities:

Your Turn

42

932

yx

yx

2

3

21

32

xy

xy

You work two jobs and can work no more than a total of 25 hours per week. You make $8/hr at the first job and $10/hr at the second job. Your boss at the second job can only give you 10 hours each week. Write and graph a system of inequalities assuming that you need to earn at least $150/week.

Your Turn

Two Jobs Problem

10

25

158.

y

xy

xy

pg 136 (# 1-17 odd)

Homework

Practice 3-3 WS (# 1-13 odd)

Homework

Systems of Three Equations and Three Variables

One Day

Solving Systems of Three Equations Algebraically

1. When we have three equations in a system, we can use the same two methods to solve them algebraically as with two equations.

2. Whether you use substitution or elimination, you should begin by numbering the equations!

Solving Systems of Three Equations

Linear Combination Method1. Choose two of the equations and eliminate

one variable as before.2. Now choose one of the equations from step 1

and the other equation you didn’t use and eliminate the same variable.

3. You should now have two equations (one from step 1 and one from step 2) that you can solve by elimination.

4. Find the third variable by substituting the two known values into any equation.

Example

1934

1532

433

zyx

zyx

zyx

Example

144

113

42

zyx

zyx

zyx

Example

42

42

33

zyx

zyx

zyx

pg 157 (# 1-9 odd, 10)

Homework

Solving Systems of Three Equations

Substitution Method 1. Choose one of the three equations and

isolate one of the variables.

2. Substitute the new expression into each of the other two equations.

3. These two equations now have the same two variables. Solve this 2 x 2 system as before.

4. Find the third variable by substituting the two known values into any equation.