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Solving Systems of Equations and Inequalities

- Section 2.1 Two variable linear

equations - Section 2.2 Three variable linear

equations - Section 2.3 Matrices Resolution of linear

systems - Section 2.5 Determinants Inverses of Matrices
- Section 2.6 Solving Systems of Linear

Inequalities - Section 2.7 Linear

Programming

Section 2.1 Two Variable Linear Equations

- A set of two or more linear equations that each

contain two variables. - These equations represent lines that will

intersect, overlap, or be parallel to each other.

Section 2.1 Two Variable Linear Equations

- When these lines intersect or overlap each other

they are said to be consistent. This means they

have at least one point of intersection. If

there is only one point of intersection, the

lines are independent. If there is more than one

point of intersection, the lines are dependent. - When two lines do not intersect, they are said to

be inconsistent because the lines are parallel.

Section 2.1 Two Variable Linear Equations

- Check your understanding
- Given the following two linear equations,

determine the consistency dependence of the

system. 2x y 5 4x 2y 8 - Consistent, independent
- Consistent, dependent
- Inconsistent

Section 2.1 Two Variable Linear Equations

- Check your understanding
- Given the following two linear equations,

determine the consistency dependence of the

system. 2x y 5 4x 2y 8 - Consistent, independent
- Consistent, dependent
- Inconsistent

Section 2.1 Two Variable Linear Equations

- Check your understanding
- Given the following two linear equations,

determine the consistency dependence of the

system. 3x 2y 5 6x 2y 8 - Consistent, independent
- Consistent, dependent
- Inconsistent

Section 2.1 Two Variable Linear Equations

- Check your understanding
- Given the following two linear equations,

determine the consistency dependence of the

system. 3x 2y 5 6x 2y 8 - Consistent, independent
- Consistent, dependent
- Inconsistent

Section 2.1 Two Variable Linear Equations

- Systems of two linear equations are usually

solved using either the elimination method or the

substitution method. - The elimination method uses a process of removing

one of the two variables simultaneously from both

equations through addition of equal but opposite

coefficients.

In blue we have y 3x - 2, and in red we have y

-x 2.

Section 2.1 Two Variable Linear Equations

In blue we have y 3x - 2, and in red we have y

-x 2.

Blue 3x y 2 Red x y 2

4x 4

With the variable y having equal but opposite

values we can add the two equations together and

eliminate the y variable. We get 4x 4. This

allows x 1, and by inserting x 1 back into

the equation we can see that y 1.

Section 2.1 Two Variable Linear Equations

- The substitution method uses a process of

rewriting one of the two equations by isolating

one of the variables and then substituting the

equation into the other equation.

In blue we have y 3x - 2, and in red we have y

-x 2.

With both equations in the form of y equals, we

can substitute the 2nd equation into the first

and we get -x 2 3x 2. Solving for x we

get -4x -4 and x 1.

Section 2.1 Two Variable Linear Equations

- Check your understanding
- Given the following two linear equations, use the

substitution method to start the solution of the

system. - 5x 3y 22 6x 2y 20
- Y 10 6x
- Y 10 3x
- Y 10 3x

Section 2.1 Two Variable Linear Equations

- Check your understanding
- Given the following two linear equations, use the

substitution method to start the solution of the

system. - 5x 3y 22 6x 2y 20
- Y 10 6x
- Y 10 3x
- Y 10 3x

Section 2.2 Three variable linear systems

- Three linear equation systems are solved using

the elimination and substitution methods. - The technique requires the isolation of one of

the variables amongst the three equations. - This gets repeated for a 2nd variable and the

remaining variable is then determined. - Afterwards, the other variables are determined.

Section 2.2 Three variable linear systems

- Sample
- X 2y 2z 10
- 2x y 2z 6
- X 3y 2z 1
- Solve for x, y, z.

Isolate the z value first. Combine line 1 2

and then combine line 3 2.

X 2y 2z 10 ? X 2y 2z 10 2x y 2z

6? -2x y 2z -6

-x 3y 4

X 3y 2z 1 ? X 3y 2z 1 2x y

2z 6 ? -2x y 2z -6

-x - 2y -5

Section 2.2 Three variable linear systems

- Sample
- X 2y 2z 10
- 2x y 2z 6
- X 3y 2z 1
- Solve for x, y, z.

Isolate the x value next. Combine line 1 2

answer and the line 3 2 answer.

X 2y 2z 11 ? X 2y 2z 11 2x y 2z

6 ? -2x y 2z -6

-x 3y 5

X 3y 2z 1 ? X 3y 2z 1 2x y

2z 6 ? -2x y 2z -6

-x - 2y -5

-x 3y 5 ? -x 3y 5 -x -2y -5 ? x 2y

5 5y 10 Y 2, x 1

Solve for z by reinserting y x values. 1

2(2) 2z 10 Z 2.5

Section 2.3 - Matrices

- Matrix A rectangle array of terms (elements)

arranged in columns and rows. A matrix with m

rows and n columns is called an m x n matrix,

(read m by n matrix). - Matrices are also used to determine solutions for

multiple variable linear equations. This

technique can be used as an alternative to

elimination or substitution methods.

Section 2.3 - Matrices

- a11 a12 a13
- a21 a22 a23
- a31 a32 a33

3 x 3 Matrix The first number indicates the row

(horizontal) and the second number indicates the

column number (vertical).

Equal Matrices Two matrices are equal if and

only they have the same dimensions and are equal

element by element.

This expression states that Y 2x 6 and x

2y. Using the substitution method, we see that

Y 2(2y) 6 and so y 2, x 4.

2x 6 2y

Y X

Section 2.3 - Matrices

Addition of Matrices The sum of two m x n

matrices is a m x n matrix in which the elements

are the sum of the corresponding elements of the

given matrices.

-6 7 -1 4 -3 10

Solve for A B.

-2 0 1 0 5 -8

B

A

-2 (-6) 0 7 1 (-1) 0 4 5 (-3)

-8 10

-8 7 0 4 2 2

A B

A B

Section 2.3 - Matrices

Subtraction of Matrices The difference of two m

x n matrices is equal to the sum A (-B) where

(-B) is the additive inverse of B.

-6 7 -1 4 -3 10

Solve for A - B.

-2 0 1 0 5 -8

B

A

-2 - (-6) 0 - 7 1 - (-1) 0 - 4 5 - (-3)

-8 - 10

4 -7 2 -4 8 -18

A - B

A - B

Section 2.3 - Matrices

Scalar Product The product of a scalar k and an

m x n matrix A is an m x n matrix denoted by kA.

Each element of kA equals k times the

corresponding element of A.

Solve for kA.

5

-2 0 1 0 5 -8

k

A

5(-2) 5(0 ) 5(1) 5(0) 5(5) 5(-8)

-10 0 5 0 25 -40

kA

kA

Section 2.5 Determinants and Inverses

- A determinant is a square array of numbers

(written within a pair of vertical lines) which

represents a certain sum of products. - Calculating a 2 2 Determinant
- In general, we find the value of a 2 2

determinant with elements a, b, c, d as follows - We multiply the diagonals (top left bottom

right first), (bottom left x top right) then

subtract the first product minus the second.

Cramers Rule

Cramers Rule begins with the solving of the

determinant for the system followed by the

determinants for each of the variables within the

system.

The determinant for each of the variables is

calculated by first substituting the solution

column values for the variable column values and

then worked on as a 2 x 2 matrix.

Cramers Rule (continued)

Cramers Rule (conclusion)