Review 2.1-2.3

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Review 2.1-2.3
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Ex Check whether the ordered pairs are
solutions of the system.x-3y -5-2x3y10

• (-5,0)
• -5-3(0) -5
• -5 -5
• -2(-5)3(0)10
• 1010
• Solution

• (1,4)
• 1-3(4) -5
• 1-12 -5
• -11 -5
• doesnt work in the 1st eqn, no need to check
  the 2nd.
• Not a solution.

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Solving a System Graphically

• Graph each equation on the same coordinate plane.
  (USE GRAPH PAPER!!!)
• If the lines intersect The point (ordered pair)
  where the lines intersect is the solution.
• If the lines do not intersect
• They are the same line infinitely many
  solutions (they have every point in common).
• They are parallel lines no solution (they share
  no common points).

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Ex Solve the system graphically.2x-2y
-82x2y4
(-1,3)
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Ex Solve the system graphically.2x4y12x2y6

• 1st eqn
• y -½x 3
• 2ND eqn
• y -½x 3
• What does this mean?
• the 2 equations are for the same line!
• Infinite many solutions

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Ex Solve graphically x-y5 2x-2y9

• 1st eqn
• y x 5
• 2nd eqn
• y x 9/2
• What do you notice about the lines?
• They are parallel!
• No solution!

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Solving Systems of Equations using Substitution
Steps 1. Solve one equation for one
variable (y x a) 2. Substitute the
expression from step one into the other
equation. Then solve. 3. Substitute back into
Step 1 and solve for the other variable. 4.
Check the solution in both equations of the
system.
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1) Solve the system using substitution

• x y 5
• y 3 x

Step 1 Solve an equation for one variable.
The second equation is already solved for y!
Step 2 Substitute
x y 5x (3 x) 5
2x 3 5 2x 2 x 1
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1) Solve the system using substitution

• x y 5
• y 3 x

y 3 x Y 3 (1) y 4
Step 3 Plug back in to find the other variable.
(1, 4) (1) (4) 5 (4) 3 (1)
Step 4 Check your solution.
The solution is (1, 4). What do you think the
answer would be if you graphed the two equations?
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Which answer checks correctly?
3x y 4 x 4y - 17

1. (2, 2)
2. (5, 3)
3. (3, 5)
4. (3, -5)

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2) Solve the system using substitution

• 3y x 7
• 4x 2y 0

It is easiest to solve the first equation for
x. 3y x 7 -3y -3y x -3y 7
Step 1 Solve an equation for one variable.
Step 2 Substitute
4x 2y 0 4(-3y 7) 2y 0
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2) Solve the system using substitution

• 3y x 7
• 4x 2y 0

-12y 28 2y 0 -14y 28 0 -14y -28 y 2
x -3y 7 x -3(2) 7 x -6 7 x 1
Step 3 Plug back in to find the other variable.
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2) Solve the system using substitution

• 3y x 7
• 4x 2y 0

Step 4 Check your solution.
(1, 2) 3(2) (1) 7 4(1) 2(2) 0
When is solving systems by substitution easier to
do than graphing? When only one of the equations
has a variable already isolated (like in example
1).
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If you solved the first equation for x, what
would be substituted into the bottom equation.
2x 4y 4 3x 2y 22

1. -4y 4
2. -2y 2
3. -2x 4
4. -2y 22

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3) Solve the system using substitution

• x 3 y
• x y 7

Step 1 Solve an equation for one variable.
The first equation is already solved for x!
Step 2 Substitute
x y 7 (3 y) y 7
3 7 The variables were eliminated!! This is a
special case. Does 3 7? FALSE!
When the result is FALSE, the answer is NO
SOLUTIONS.
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3) Solve the system using substitution

• 2x y 4
• 4x 2y 8

Step 1 Solve an equation for one variable.
The first equation is easiest to solved for y! y
-2x 4
4x 2y 8 4x 2(-2x 4) 8
Step 2 Substitute
4x 4x 8 8 8 8 This is also a special
case. Does 8 8? TRUE!
When the result is TRUE, the answer is INFINITELY
MANY SOLUTIONS.
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What does it mean if the result is TRUE?

1. The lines intersect
2. The lines are parallel
3. The lines are coinciding
4. The lines reciprocate
5. I can spell my name

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Solving Systems of Equations using
Elimination
Steps 1. Place both equations in Standard Form
Ax By C. 2. Determine which variable to
eliminate with Addition or Subtraction. 3.
Solve for the variable left. 4. Go back and use
the found variable in step 3 to find second
variable. 5. Check the solution!!!!
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1) Solve the system using elimination.

• 2x 2y 6
• 3x y 5

Step 1 Put the equations in Standard Form.
They already are!
None of the coefficients are the same! Find
the least common multiple of each variable.
LCM 6x, LCM 2y Which is easier to
obtain? 2y(you only have to multiplythe bottom
equation by 2)
Step 2 Determine which variable to eliminate.
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1) Solve the system using elimination.
2x 2y 6 3x y 5
Multiply the bottom equation by 2 2x 2y
6 (2)(3x y 5)
8x 16 x 2
2x 2y 6 () 6x 2y 10
Step 3 Multiply the equations and solve.
2(2) 2y 6 4 2y 6 2y 2 y 1
Step 4 Plug back in to find the other variable.
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1) Solve the system using elimination.
2x 2y 6 3x y 5
(2, 1) 2(2) 2(1) 6 3(2) - (1) 5
Step 5 Check your solution.
Solving with multiplication adds one more step to
the elimination process.
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2) Solve the system using elimination.

• x 4y 7
• 4x 3y 9

Step 1 Put the equations in Standard Form.
They already are!
Find the least common multiple of each
variable. LCM 4x, LCM 12y Which is easier
to obtain? 4x(you only have to multiplythe top
equation by -4 to make them inverses)
Step 2 Determine which variable to eliminate.
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2) Solve the system using elimination.

• x 4y 7
• 4x 3y 9

Multiply the top equation by -4 (-4)(x 4y
7) 4x 3y 9)
y 1
-4x 16y -28 () 4x 3y 9
Step 3 Multiply the equations and solve.
-19y -19
x 4(1) 7 x 4 7 x 3
Step 4 Plug back in to find the other variable.
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2) Solve the system using elimination.
x 4y 7 4x 3y 9
(3, 1) (3) 4(1) 7 4(3) - 3(1) 9
Step 5 Check your solution.
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What is the first step when solving with
elimination?

1. Add or subtract the equations.
2. Multiply the equations.
3. Plug numbers into the equation.
4. Solve for a variable.
5. Check your answer.
6. Determine which variable to eliminate.
7. Put the equations in standard form.

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Which variable is easier to eliminate?
3x y 4 4x 4y 6

1. x
2. y
3. 6
4. 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32
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3) Solve the system using elimination.

• 3x 4y -1
• 4x 3y 7

Step 1 Put the equations in Standard Form.
They already are!
Find the least common multiple of each
variable. LCM 12x, LCM 12y Which is easier
to obtain? Either! Ill pick y because the signs
are already opposite.
Step 2 Determine which variable to eliminate.
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3) Solve the system using elimination.

• 3x 4y -1
• 4x 3y 7

Multiply both equations (3)(3x 4y
-1) (4)(4x 3y 7)
x 1
9x 12y -3 () 16x 12y 28
Step 3 Multiply the equations and solve.
25x 25
3(1) 4y -1 3 4y -1 4y -4 y -1
Step 4 Plug back in to find the other variable.
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3) Solve the system using elimination.
3x 4y -1 4x 3y 7
(1, -1) 3(1) 4(-1) -1 4(1) - 3(-1) 7
Step 5 Check your solution.
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What is the best number to multiply the top
equation by to eliminate the xs?
3x y 4 6x 4y 6

1. -4
2. -2
3. 2
4. 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32
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Solve using elimination.
2x 3y 1 x 2y -3

1. (2, 1)
2. (1, -2)
3. (5, 3)
4. (-1, -1)

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Find two numbers whose sum is 18 and whose
difference 22.

1. 14 and 4
2. 20 and -2
3. 24 and -6
4. 30 and 8

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Using Elimination to Solve a Word Problem
Two angles are supplementary. The measure of one
angle is 10 degrees more than three times the
other. Find the measure of each angle.
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Using Elimination to Solve a Word Problem
Two angles are supplementary. The measure of one
angle is 10 more than three times the other. Find
the measure of each angle.
x degree measure of angle 1 y degree
measure of angle 2 Therefore x y 180
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Using Elimination to Solve a Word Problem
Two angles are supplementary. The measure of one
angle is 10 more than three times the other. Find
the measure of each angle.
x y 180
x 10 3y
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Using Elimination to Solve a Word Problem
Solve x y 180
x 10 3y
x 42.5 180 x
180 - 42.5 x 137.5 (137.5, 42.5)
x y 180 -(x - 3y 10) 4y 170
y 42.5
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Using Elimination to Solve a Word Problem
The sum of two numbers is 70 and their difference
is 24. Find the two numbers.
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Using Elimination to Solve a Word problem
The sum of two numbers is 70 and their difference
is 24. Find the two numbers.
x first number y second number Therefore, x
y 70
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Using Elimination to Solve a Word Problem
The sum of two numbers is 70 and their difference
is 24. Find the two numbers.
x y 70
x y 24
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Using Elimination to Solve a Word Problem
x y 70 x - y 24
47 y 70 y 70 47 y 23
2x 94 x 47
(47, 23)
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Now you Try to Solve These Problems Using
Elimination.
Solve

1. Find two numbers whose sum is 18 and whose
   difference is 22.

1. The sum of two numbers is 128 and their
   difference is 114. Find the numbers.

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Matrix Operations
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What is a Matrix?

• MATRIX A rectangular arrangement of numbers in
  rows and columns.
• The ORDER of a matrix is the number of the rows
  and columns.
• The ENTRIES are the numbers in the matrix.

• This order of this matrix is a 2 x 3.

columns
rows
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What is the order?
(or square matrix)
3 x 3
(Also called a column matrix)
1 x 4
3 x 5
(or square matrix)
2 x 2
4 x 1
(Also called a row matrix)
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Adding Two Matrices

• To add two matrices, they must have the same
  order. To add, you simply add corresponding
  entries.

46

7
7
4
5

0
7
5
7
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Subtracting Two Matrices

• To subtract two matrices, they must have the same
  order. You simply subtract corresponding entries.

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-4-1
3-8
2-0
-5
2
-5
8-3
0-(-1)
-7-1
5
-8
1


5-2
1-(-4)
0-7
5
3
-7
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Multiplying a Matrix by a Scalar

• In matrix algebra, a real number is often called
  a SCALAR. To multiply a matrix by a scalar, you
  multiply each entry in the matrix by that scalar.

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-3
3
-2
6
-5
-2(-3)
-2(3)
6
-6
-12
-2(6)
-2(-5)
10
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Multiplying Matrices
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In order to multiply matrices...
A B AB
m x n
n x p
m x p
Ex 1.
Can you multiply? What will the dimensions be?
A
B
AB
2 x 3
3 x 4
2 x 4
AB
A
B
5 x 3
5 x 2
Not possible
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How to multiply...
ac
ad

2 x 1
1 x 2
2 x 2
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How to multiply...
ac
ad

bc
bd
2 x 1
1 x 2
2 x 2
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Ex. 1
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Ex. 2
Find AB
-16
1
-12
2
-15
-10

0
- 4
0
- 2
- 4
-2
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Ex. 3
Find BA
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How to multiply...
ag
bi
ck
ah
bj
cm

2 x 3
3 x 2
2 x 2
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How to multiply...
ag
bi
ck
ah
bj
cm

dg
ei
fk
dh
ej
fm
2 x 3
3 x 2
2 x 2
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Ex. 4
-1(4)
5(6)
-1(-3)
5(8)
-4
30
3
40
-15
16
20
12
5(-3)
2(8)
5(4)
2(6)

0
-24
0
-32
0(4)
-4(6)
0(-3)
-4(8)
26
43
32
1
-24
-32
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If we are multiplying matrices, we multiply each
row of the first matrix by each column in the
second matrix!!
multiply each row of the first matrix by each
column in the second matrix!!


1


2


1


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2 X 3
3 X 3
2(3)(-3)(8)5(-3)
2(9)(-3)(0) 5(1)
2(1)(-3)(-4) 5(5)
-1(3)(6)(8)8(-3)
-1(9)(6)(0) 8(1)
-1(1)(6)(-4) 8(5)
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Ex. 5
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Examples
2(3) -1(5)
2(-9) -1(7)
2(2) -1(-6)
3(-9) 4(7)
3(2) 4(-6)
3(3) 4(5)
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Dimensions 2 x 3 2 x 2
They dont match so cant be multiplied
together.
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Answer should be a 2 x 2
0(4) (-1)(-2)
0(-3) (-1)(5)
1(4) 0(-2)
1(-3) 0(5)
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Solving Systems of Equationswith Matrices
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A system of equations may be represented as a
matrix equation. For example, the system of
equations
may be represented by the matrix equation
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Write the matrix equation that represents the
system
70
Write the matrix equation that represents the
system
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A matrix equation is in the form AX B, where A
is the coefficient matrix, X is the variable
matrix, and B is the constant matrix.
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Solving AXB
Real Numbers axb (1/a)(ax) (1/a)b (?1/a)(a)x b/a 1x b/a x b/a
Note 1/a must exist to solve ax b
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Solving AXB
Real Numbers axb (1/a)(ax) (1/a)b (?1/a)(a)x b/a 1x b/a x b/a Matrices AXB A-1(AX)A-1B (A-1A)XA-1B IXA-1B XA-1B
Note A-1 must exist to solve AXB
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Solve the system of equations using matrices.
x -7 y 15
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Solve the system of equations using matrices.
x 1 y -1
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Ex. 2 Solve using matrices.
AX B
X A-1B
x -7 y -4
B
A
(-7, -4)
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Ex. 3 Solve using matrices
x 5/7 y 2
(5/7, 2)