Notes for Homework 8, due March 28

Notes on determinants

A 2x2 matrix is an array of four numbers put in two rows and two columns, such as

| a c |
| b d |

The determinant of this matrix is ad - bc, the product of the main diagonal minus the product of the opposite diagonal. If you use Kramer's rule to solve a pair of simultaneous equations, you will need to know how to calculate determinants.

For example, let's consider the following set of equations.

3x - 2y = 7
2x + 4y = 12

As an augmented matrix , these equations become

|  3  -2  : 7 |
|  2   4  : 12|

Matrix_1 is

| 3 -2 |
| 2  4 |

The determinant is 3(4) - 2(-2) = 12 - -4 = 16
 

Matrix_x is

| 7 -2 |
| 12  4 |

The determinant is 7(4) - 12(-2) = 28 - -24 = 52
  Matrix_y is

| 3  7  |
| 2 12 |

The determinant is 3(12) - 2(7) = 36 - 14 = 22

From here, Kramer's rule then has x = 52/16 = 13/4 or 3 1/4. The value for y = 22/16 = 11/8 = 1 3/8
 
  Notes on solving simultaneous equations
 

Simultaneous equation practice with Kramer's rule

6x + 5y = 18
2x - y = -4

3x + 7y = 12
2x + 5y = -18

Answers in the comments.

Notes for Homework 9

Slope, point-slope, slope-intercept

Solving simultaneous equations

Solving simultaneous equations: Elimination

Solving simultaneous equations: Kramer's Rule 

Answers to homework #10

When F = 40, C = 5/9(40-32) = 5/9(8) = 4.44 degrees

When F = 60, C = 5/9(60-32) = 5/9(28) = 15.56 degrees

When C = 30, F = 9/5(30) + 32 = 9*6 + 32 = 54 + 32 = 86 degrees

When C = 40, F = 9/5(40) + 32 = 9*8 + 32 = 72 + 32 = 104 degrees

Find where y = 7x + 4 and y = 10x - 5 intersect.

Step 1: Set the equations equal to each other and solve for x.

7x + 4 = 10x - 5_____Subtract 7x from both sides.

4 = 3x - 5_________Add 5 to both sides.

9 = 3x___________Divide both sides by 3.

3 = x

Step 2: Plug x=3 into either equation to find y.

y = 7(3) + 4 = 25

The point of intersection is (3, 25).


Kramer's rule problems.

3x + 4y = 25
4x - 2y = 4

Matrix version
[3 _4 | 25]
[4 -2 | _4]

Three matrices.

M
[3 _4 ]
[4 -2 ]

determinant = 3x(-2) - 4x4 = -6 - 16 = -22

Mx
[25 _4 ]
[ 4 -2 ]

determinant = 25x(-2) - 4x4 = -50 - 16 = -66

My
[3 _25 ]
[4 __4 ]

determinant = 3x(4) - 4x25 = 12 - 100 = -88

x = -66/-22 = 3
y = -88/-22 = 4

Answer: (3, 4)

2x + 4y = 10
4x + 8y = 30

Matrix version
[2 4 | 10]
[4 8 | 30]

Three matrices.

M
[2 4 ]
[4 8 ]

determinant = 2x8 - 4x4 = 16 - 16 = 0. This means no single solution.

Mx
[10 4 ]
[30 8 ]

determinant = 10x8 - 30x4 = 80 - 120 = -40. Since this isn't 0, the two lines are parallel.

We don't have to do the third determinant. Yay!

Answer: No solution, two parallel lines.

_x + _y = 13
3x - 3y = 9

Matrix version
[1 _1 | 13]
[3 -3 | _9]

Three matrices.

M
[1 _1 ]
[3 -3 ]

determinant = 1x(-3) - 3x1 = -3 - 3 = -6

Mx
[13 1 ]
[ 9 -3 ]

determinant = 13x(-3) - 9x1 = -39 - 9 = -48

My
[1 _13 ]
[3 __9 ]

determinant = 1x9 - 3x13 = 9 - 39 = -30

x = -48/-6 = 8
y = -30/-6 = 5

Answer: (8, 5)

Notes for Homework 9, due Monday, Oct. 30

Notes on simultaneous equations and Kramer's Rule