Algebra 213.3a

Niagara Falls

Get ready for a “Small Quiz” to be written

on your grade sheet.

Quiz. Copy the problems and write the answer.

Put your grade paper on the front of your row, quiz side down.

1. Solve using substitution:

2. Solve using elimination:

3 8

4 2 6

x y

x y

3 4 4

5 3 26

x y

x y

3 4 5

1 2 6

A rectangular array of numbers is called a matrix. Example:

A

This is a 2×3 matrix (2 rows and 3 columns).Every square matrix is associated with a number called its determinant. The determinant of matrix A is denoted , (not absolute value).

a b

c d

a b

c d

Definition: The determinant of matrix is:

(You start at the upper left, just like reading a book.)

ad bc

1. Evaluate: 7 3

4 5 35 12 23

2. Evaluate: 1 2

3 4

4 6 10

3. Evaluate: 7 2

1 3

21 2 23

4. Evaluate: 2 3

4 x

2 12x

Solve using Cramer’s RuleTo solve using Cramer’s Rule, we need one more determinant than we have equations. The first determinant is called the denominator determinant, and it is made from the coefficients of x and y.

Example: If we solve5 4

3 35

11

2

x y

x y

then the denominator determinant 5 4

3 5D

25 12 37

To find x, we replace the numbers on the right side of each equation with the numbers in the x column. So the numerator determinant for x is

455 ( 92) 37

11

23 5xD

So the value of x is 37

137

xD

D

In the same way, in making the yD numerator determinant,

we replace the y column with the answer column:

5 11115 33 148

3 23yD

5 4

3 35

11

2

x y

x y

The value of y is 1484

37yD

D

So the answer is (1, 4).

In summary, to find the solution to a system of equations by using Cramer’s Rule, we find:

and yxDD

D D

.

Three equations: If we have three equations with 3 unknowns, we find

and andyx zDD D

D D D

.

3 7

6 4 9

x y

x y

You work:

; ;x yDDD

x y

3 7

6 4 9

x y

x y

You work:

1 34 ( 18) 22

6 4D

7 328 ( 27) 55

9 4xD

1 79 42 33

6 9yD

55 5

22 2xDxD

33 3

22 2yDyD

5 3,

2 2

Classwork: 2, 10/580