Today:

Warm-Up: (3)

Review Systems Solutions: (6)

Solving 3x3 Systems:(2)

Complete Class Work from Yesterday

Warm Up

1. Write an equation for a line perpendicular to 2x -4y = -2

2. Solve for a: 9a – 2b = c + 4a

a. Write the equation of the lineb. Write the inequality of the graph.

4. Write the systems of equations shown by the graph below.

Graphing to Solve a Linear System

Step 1: Put both equations in slope - intercept form.

Step 2: Graph both equations on

the same coordinate plane.

Step 3: Plot the point where the graphs intersect.

Step 4: Check to make sure your

solution makes both equations true.

Solve both equations for y, so that

each equation looks like

y = mx + b.

Use the slope and y - intercept for each equation in step 1.

This is the solution! LABEL the solution!

Substitute the x and y values into

both equations to verify the point is

a solution to both equations.

4-Step Summary

Review: Solve Systems of Equations by Graphing: (1)

( + 0)

Step 1: Put both equations in slope - intercept form.

Step 2: Graph both equations on

the same coordinate plane.

Step 3: Plot the point where the graphs intersect.

(2,1)

Step 4: Check to make sure your

solution makes both equations true.

1 = 1 + 0

2 + 1 = 3

1: Put the equations in Standard Form.

Step 2: Determine which variable to eliminate.

Step 3: Add or subtract the equations.

Step 4: Plug back in to find other variable.

Step 5: Check your solution.

Standard Form: Ax + By = C

Look for variables that have the same coefficient.

Solve for the variable.

Substitute the value of the variable into the equation.

Substitute your ordered pair

into BOTH equations.

Elimination is easiest when the equations are in standard form.

Solve Systems of Equations by Elimination (add. or subtract)

2x + 7y = 31

5x - 7y = - 45

7x + 0 = -14 x = -2

THEN----

Like variables must be lined under each other.

Solve Systems of Equations by Elimination:(2) (addition or subtraction)

2x + 7y = 31

(-2, 5)

Substitute your answer into either original equation and solve for the second variable.

Solution

Now check our answers in both equations------

2(-2) + 7y = 31-4 + 7y = 314 4

7y = 35; y = 5

Review: Solve Systems of Equations by Elimination (addition or subtraction)

2x + 7y = 31

2(-2) + 7(5) = 31

-4 + 35 = 31

31 = 31

5x – 7y = - 45

5(-2) - 7(5) = - 45

-10 - 35 = - 45

- 45 =- 45

What variable to eliminate?

Like variables must be lined under each other.

x + y = 4

2x + 3y = 9

1x + 1y = 4

2x + 3y = 9

Solve Systems of Equations by Elimination(multiplying)

By doing what?

2x + 3y = 9

-2x - 2 y = - 8

2x + 3y = 9

Now add the two equations and solve.

y = 1

THEN----

x + y = 4( ) -2

Solve Systems of Equations by Elimination (multiplying)

(3,1)

Substitute your answer into either original equation and solve for the second variable.

Solution

Now check our answers in both equations--

x + y = 4

x + 1 = 4

- 1 -1

x = 3

Solve Systems of Equations by Elimination (multiplying)

x + y = 43 + 1 = 4

4 = 4

2x + 3y = 9

2(3) + 3(1) = 9

6 + 3 = 9

9 = 9

Can you multiply either equation by an integer in order to eliminate

one of the variables?

Here, we must multiply bothequations by a (different) number in order to easily

eliminate one of the variables.

Multiply the top equation by 2, and the bottom equation by -3

Eliminate Plug back in Solve for other

variable

3x – 2y = -7

2x -5y = 10

Write your solution as an ordered pair

(-5,-4)

Plug both solutions into original equations

Solve Systems of Equations by Elimination (multiplying)

3x – 2y = -7-15 – (-8) = -7

-7 = - 7

2x - 5y = 10

-10 – (-20) = 10

10= 10

Solving a system of equations by substitution

Step 1: Solve an equation for one variable.

Step 2: Substitute

Step 3: Solve the equation.

Step 4: Plug back in to find

the other variable.

Step 5: Check your

solution.

Pick the easier equation. The goal

is to get y= ; x= ; a= ; etc.

Put the equation solved in Step 1

into the other equation.

Get the variable by itself.

Substitute the value of the variable

into the equation.

Substitute your ordered pair into

BOTH equations.

Solve: By Substitution: (2)

The same way, the substitution method is very closely related to the elimination method.

After eliminating one variable and solving for the other, we substitute the value of the variable back into the equation.

2x + 3y = -26 4x - 3y = 2

What is the value of x ?

At this point we substitute -4 for x, and solve for y. This is exactly what the substitution method is except it is done at the beginning.

-4

For example: Find the value of ‘x’ using elimination.

Solve By Substitution: (2)

Example 1: 2x + y = 04x - y = -4

Solve for y in the first equation

Example 2: y = 2x - 16x - 3y = 7

Substitution won’t always be so obvious. You may have to rearrange the equation to isolate the variable.

Solving 3x3 Systems:(2)The graph of the solution set of an equation in threevariables is a plane, not a line. In fact, graphing equations in three variables requires the use of a three-dimensional coordinate system. It is therefore, not practical to solve

3x3 systems by graphing

Solve the System: 4x + 2y - z = -5 3y + z = -1

2z = 10

1. Which is the easiestvariable to solve for?

2. Plug in where and solve for what?

3. Substitute y and z values into equation 1; solve for x.

4. Substitute all values, check for equality.5. The solution set is (1, -2, 5)

Solving 3x3 Systems: (2)

Practice 2: -3x - 2y - z = -17 3x + 5y = 29-5x = -15

Solving 3x3 Systems: (2)

5. A restaurant charged one customer $28.20 for 3 small dishes and 5 large dishes and charged another customer $23.30 for 4 small dishes and 3 large dishes. What will 2 small and 4 large dishes cost?

A shopper bought two pairs of gloves and four hats for $42.00. Her friend bought two pairs of gloves and two hats for $30.00. What is the price of each item?

Use ‘g’ for gloves and ‘h’ for hats.

Equation 1: 2g + 4h = $42.00Equation 2: 2g + 2h = $30.00

0g + 2h = $12.00

Hats are $6.00 each

2g + 24 = $42.00

Plug both values in and check for

equality.

Systems of Equations Word Problems: (2)

Gloves are $9.00 ea.

For Valentines Day Mark bought his mom 12 flowers, a mixture of roses and daisies. The roses cost $1.15 each and the daisies cost $1.35 each. If he spent $16.00, how many daisies did he buy?

x =

y =

# of

roses

# of

daisies

12x y

1.15 1.35 16.00x y 100 100 100

115x 135y 1600115x

12x y

12x 1111 11

1x

11 daisies

1. Mark

the text.

2. Label

variables.

3. Create

equations.

4. Solve.

5. Check.

115 115 115

115y 1380

20 20

11y

20 220y

Let’s eliminate

the ‘x’

Try solving by elimination

Complete all class work problems