Warm-Up

Find the x, y and z intercepts:

a) 3𝑥 + 4𝑦 + 6𝑧 = 24

b) 2𝑥 + 5𝑦 + 10𝑧 = 10

Solve this 2-D system by Graphing on your

calculator

c) −2𝑥 + 3𝑦 = 45

4𝑥 + 5𝑦 = 10

Solving Systems of

Equations

Learning Targets

Refresher on solving systems of equations

Matrices

– Operations

– Uses

– Reduced Row Echelon Form

Solving Systems of Equations

There are multiple ways to solve systems of

equations:

– Graphing

– Substitution (Equal Values Method)

– Elimination

Solve the System by Graphing

3𝑦 − 2𝑥 = 45

5𝑦 + 4𝑥 = 10

f(x)=(2/3)x+15

f(x)=-(4/5)x+2

Series 1

-19-18-17-16-15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1

123456789

10111213141516171819

x

y

(-8.86,9.09)

Solve the System using Algebra

4𝑥 + 3𝑦 = 12

2𝑥 + 2𝑦 = 14

Algebra Method cont.

Elimination Method:

4𝑥 + 3𝑦 = 12

2𝑥 + 2𝑦 = 14 4𝑥 + 3𝑦 = 12

−2(2𝑥 + 2𝑦 = 14)

4𝑥 + 3𝑦 = 12

−4𝑥 − 4𝑦 = −28

−𝑦 = −16

𝑦 = 16

2𝑥 + 2 16 = 14

2𝑥 + 32 = 14

2𝑥 = −18

𝑥 = −9

Matrix Equations

We have solved systems

using graphing, but now

we learn how to do it

using matrices. This will

be particularly useful

when we have equations

with three variables.

Matrix Equation

Before you start, make sure:

1. That all of your equations are in

standard form.

2. The variables are in the same

order (alphabetical usually is

best).

3. If a variable is missing use zero

for its coefficient.

Setting up the Matrix

Equation

Given a system of equations

-2x - 6y = 0

3x + 11y = 4

Since there are 2 equations,

there will be 2 rows.

Since there are 2 variables,

there will be 2 columns.

There are 3 parts to a matrix

equation

1)The coefficient matrix,

2)the variable matrix, and

3)the constant matrix.

Setting up the Matrix

Equation

-2x - 6y = 0

3x + 11y = 4

The coefficients are placed

into the coefficient matrix.

2 6

3 11

-2x - 6y = 0

3x + 11y = 4

Your variable matrix will

consist of a column.

x

y

-2x - 6y = 0

3x + 11y = 4

The matrices are multiplied

and represent the left side

of our matrix equation.

x

y

2 6

3 11

-2x - 6y = 0

3x + 11y = 4

The right side consists of

our constants. Two

equations = two rows.

0

4

-2x - 6y = 0

3x + 11y = 4

Now put them together.

2 6

3 11

x

y

0

4

We’ll solve it later!

Create a matrix equation

3x - 2y = 7

y + 4x = 8

Put them in Standard Form.

Write your equation.

3 2

4 1

x

y

7

8

3a - 5b + 2c = 9

4a + 7b + c = 3

2a - c = 12

3 5 2

4 7 1

2 0 1

a

b

c

9

3

12

Create a matrix equation

To solve matrix equations, get

the variable matrix alone on

one side.

Get rid of the coefficient

matrix by multiplying by its

inverse

Solving a matrix

equation

2 6

3 11

x

y

0

4

When solving matrix equations

we will always multiply by the

inverse matrix on the left of the

coefficient and constant matrix.

(remember commutative

property does not hold!!)

The left side of the equation

simplifies to the identity times

the variable matrix. Giving

us just the variable matrix.

x

y

2 6

3 11

10

4

2 6

3 11

12 6

3 11

x

y

2 6

3 11

10

4

Using the calculator we can

simplify the left side. The

coefficient matrix will be A

and the constant matrix will

be B. We then find A-1B.

x

y

2 6

3 11

10

4

The right side simplifies to give

us our answer.

x = -6

y = 2

You can check the systems by

graphing, substitution or

elimination.

x

y

6

2

Advantages

Basically, all you have to do

is put in the coefficient

matrix as A and the constant

matrix as B. Then find A-1B.

This will always work!!!

Solve:

Plug in the coeff. matrix as A

Put in the const. matrix as B

Calculate A-1B.

3 2

4 1

x

y

7

8

x

y

21

114

11

Solve: r - s + 3t = -8

2s - t = 15

3r + 2t = -7

1 1 3

0 2 1

3 0 2

r

s

t

8

15

7

r

s

t

3

8

1

Working with Matrices on TI-83, TI-84

Source: Mathbits

Explore: • How many matrices does your calculator have?

• Use the right arrow key to move to MATH. Scroll

down and find rref. We will use this key later.

• Use the right arrow key once more to highlight EDIT.

Step 1: Go to Matrix (above the x-1 key)

Step 2: Arrow to the right to EDIT to allow for entering the matrix.

Press ENTER

Step 3: Type in the dimensions (size) of your matrix and enter the elements (press ENTER).

Step 4: Repeat this process for

a different matrix. .

Step 5: Arrow to the right to EDIT and choose a new name.

Step 6: Type in the dimensions (size) of your matrix and enter the elements (press ENTER).

Using Matrices to Solve Systems of Equations:

• 1. (using the inverse coefficient matrix)

Write this system as a matrix equation and

solve: 3x + 5y = 7 and 6x - y = -8

• Step 1: Line up the x, y and

constant values.

• 3x + 5y = 7

6x - y = -8

• Step 2: Write as equivalent

matrices.

• Step 3: Rewrite to separate out

the variables.

Step 4: Enter the two numerical matrices in the

calculator.

Step 5: The solution is obtained by multiplying both

sides of the equation by the inverse of the matrix

which is multiplied times

the variables.

• Step 6: Go to the home screen and enter the right

side of the previous equation.

• The answer to the system, as seen on the calculator

screen,

is x = -1 and y = 2.

Method 2 • 2. (using Gauss-Jordan elimination method with

reduced row echelon form )

Solve this system of equations:

• 2x - 3y + z = -5

4x - y - 2z = -7

-x + 2z = -1

• Step 1: Line up the variables and

constants

• 2x - 3y + z = -5

4x - y - 2z = -7

-x +0y + 2z = -1

• Step 2: Write as an augmented

matrix and enter into

calculator.

• Step 3: From the home screen, choose the rref

function. [Go to

Matrix (above the x-1 key), move right→MATH,

choose B: rref]

• Step 4: Choose name of matrix

and hit ENTER

• Step 5: The answer to the system, will be the last

column on the calculator screen:

x = -3

y = -1 z = -2.

Method 2: • Case 1: Unique solution

−2𝑟 + 2𝑠 + 5𝑡 = −3

−𝑟 + 5𝑠 + 4𝑡 = −15

−𝑟 + 3𝑠 + 𝑡 = −6

Enter as a

3X4 matrix

𝑟 = −4

𝑠 = −3

𝑡 = −1

Diagonal is all ones so

there is a solution:

Method 2: • Case 2: No solution

𝑥 + 5𝑦 − 𝑧 = 21

−3𝑥 + 𝑦 − 3𝑧 = −28 5𝑥 + 𝑦 + 4𝑧 = 3

Enter as a

3X4 matrix

Last row:

0 0 0 1

No solution.

Method 2: • Case 3: Infinitely Many Solutions

−5𝑚 + 𝑛 − 2𝑝 = −22 𝑚 + 3𝑛 − 6𝑝 = 14 −6𝑚 + 2𝑛 − 4𝑝 = −24

Enter as a

3X4 matrix

Last row:

0 0 0 0 Infinitely Many Solutions

Website to Visualize the Solutions

• http://www.cpm.org/flash/technology/3dsystems.s

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For Tonight

• Intro to Matrices Worksheet