review 2.1-2.3. ex: check whether the ordered pairs are solns. of the system. x-3y= -5 -2x+3y=10...

Review 2.1-2.3Review 2.1-2.3

Ex: Check whether the ordered pairs Ex: Check whether the ordered pairs are solns. of the system.are solns. of the system.

x-3y= -5x-3y= -5-2x+3y=10-2x+3y=10

A.A. (1,4)(1,4)

1-3(4)= -51-3(4)= -5

1-12= -51-12= -5

-11 = -5-11 = -5

*doesn’t work in the 1*doesn’t work in the 1stst eqn, no need to check eqn, no need to check the 2the 2ndnd..

Not a solution.Not a solution.

B.B. (-5,0)(-5,0)

-5-3(0)= -5-5-3(0)= -5

-5 = -5-5 = -5

-2(-5)+3(0)=10-2(-5)+3(0)=10

10=1010=10

SolutionSolution

Solving a System GraphicallySolving a System Graphically

1.1. Graph each equation on the same Graph each equation on the same coordinate plane. (USE GRAPH PAPER!!!)coordinate plane. (USE GRAPH PAPER!!!)

2.2. If the lines intersect: The point (ordered If the lines intersect: The point (ordered pair) where the lines intersect is the pair) where the lines intersect is the solution.solution.

3.3. If the lines do not intersect:If the lines do not intersect:a.a. They are the same line – infinitely many They are the same line – infinitely many

solutions (they have every point in common).solutions (they have every point in common).

b.b. They are parallel lines – no solution (they They are parallel lines – no solution (they share no common points).share no common points).

Ex: Solve the system graphically.Ex: Solve the system graphically.2x-2y= -82x-2y= -82x+2y=42x+2y=4

(-1,3)

Ex: Solve the system graphically.Ex: Solve the system graphically.2x+4y=122x+4y=12

x+2y=6x+2y=6 11stst eqn eqn::

x-intx-int (6,0) (6,0)

y-inty-int (0,3) (0,3) 22NDND eqn eqn::

x-intx-int (6,0) (6,0)

y-inty-int (0,3) (0,3) What does this mean?What does this mean?

the 2 eqns are for the the 2 eqns are for the same line!same line!

¸ ¸ many solutionsmany solutions

ExEx: Solve graphically: x-y=5: Solve graphically: x-y=5 2x-2y=9 2x-2y=9 11stst eqn eqn::

x-intx-int (5,0) (5,0)

y-inty-int (0,-5) (0,-5) 22ndnd eqn eqn::

x-intx-int (9/2,0) (9/2,0)

y-inty-int (0,-9/2) (0,-9/2) What do you notice What do you notice

about the lines?about the lines? They are parallel! Go They are parallel! Go

ahead, check the slopes!ahead, check the slopes! No solution!No solution!

3-2: Solving Systems of 3-2: Solving Systems of EquationsEquations

using using SubstitutionSubstitution

Solving Systems of EquationsSolving Systems of Equations using using SubstitutionSubstitution

Steps:

1. Solve one equation for one variable (y= ; x= ; a=)

2. Substitute the expression from step one into the other equation.

3. Simplify and solve the equation.

4. Substitute back into either original equation to find

the value of the other variable.

5. Check the solution in both equations of the system.

Example #1:Example #1: y = 4x3x + y = -21

Step 1: Solve one equation for one variable.

y = 4x (This equation is already solved for y.)

Step 2: Substitute the expression from step one into the other equation.

3x + y = -21

3x + 4x = -21

Step 3: Simplify and solve the equation.

7x = -21

x = -3

y = 4x3x + y = -21

Step 4: Substitute back into either original equation to find the value of the other variable.

3x + y = -21 3(-3) + y = -21 -9 + y = -21 y = -12

Solution to the system is (-3, -12).

y = 4x3x + y = -21

Step 5: Check the solution in both equations.

y = 4x

-12 = 4(-3)

-12 = -12

3x + y = -21

3(-3) + (-12) = -21

-9 + (-12) = -21

-21= -21

Solution to the system is (-3,-12).

Example #2:Example #2: x + y = 10 5x – y = 2Step 1: Solve one equation for one variable.

x + y = 10

y = -x +10Step 2: Substitute the expression from step one into

the other equation.

5x - y = 2

5x -(-x +10) = 2

x + y = 10 5x – y = 2

5x -(-x + 10) = 2

5x + x -10 = 2

6x -10 = 2

6x = 12

x = 2

Step 3: Simplify and solve the equation.

x + y = 10 5x – y = 2Step 4: Substitute back into either original

equation to find the value of the other variable.

x + y = 102 + y = 10 y = 8Solution to the system is (2,8).

x + y = 10 5x – y = 2


x + y =10

2 + 8 =10

10 =10

5x – y = 2

5(2) - (8) = 2

10 – 8 = 2

2 = 2

Solution to the system is (2, 8).

Solve by substitution:Solve by substitution:

y 2x 2

2x 3y 10

2a 3b 7

2a b 5

1.

2.

3-2: Solving Systems of 3-2: Solving Systems of EquationsEquations

using Elimination using EliminationSteps:

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.


EXAMPLE #1:EXAMPLE #1:

STEP 2: Use subtraction to eliminate 5x. 5x + 3y =11 5x + 3y = 11

-(5x - 2y =1) -5x + 2y = -1

5x + 3y = 11

5x = 2y + 1

Note: the (-) is distributed.

STEP 3: Solve for the variable. 5x + 3y =11

-5x + 2y = -15y =10 y = 2

STEP1: Write both equations in Ax + By = C form. 5x + 3y =1 5x - 2y =1

STEP 4: Solve for the other variable by substituting

into either equation.5x + 3y =11

5x + 3(2) =11 5x + 6 =11 5x = 5 x = 1

5x + 3y = 11

5x = 2y + 1

The solution to the system is (1,2).

5x + 3y= 11

5x = 2y + 1


5x + 3y = 11

5(1) + 3(2) =11

5 + 6 =11

11=11

5x = 2y + 1

5(1) = 2(2) + 1

5 = 4 + 1

5=5

The solution to the system is (1,2).

Solving Systems of EquationsSolving Systems of Equations using Elimination 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 remaining variable.

4. Go back and use the variable found in step 3 to find the second variable.


Example #2:Example #2:x + y = 10 5x – y = 2

Step 1: The equations are already in standard form: x + y = 10

5x – y = 2

Step 2: Adding the equations will eliminate y.x + y = 10 x + y = 10

+(5x – y = 2) +5x – y = +2

Step 3: Solve for the variable.x + y = 10

+5x – y = +2 6x = 12 x = 2

x + y = 10 5x – y = 2

Step 4: Solve for the other variable bysubstituting into either equation.x + y = 102 + y = 10 y = 8

Solution to the system is (2,8).

x + y = 10 5x – y = 2

x + y =10

2 + 8 =10

10=10

5x – y =2

5(2) - (8) =2

10 – 8 =2

2=2

Step 5: Check the solution in both equations.Solution to the system is (2,8).

NOW solve these using NOW solve these using

elimination:elimination:

1. 2.

2x + 4y =1

x - 4y =5

2x – y =6

x + y = 3

Using Elimination to Solve a Using Elimination to Solve a Word Problem: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.


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


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 = 180x =10 + 3y


Solvex + y = 180x =10 + 3y

x + y = 180

-(x - 3y = 10)

4y =170

y = 42.5

x + 42.5 = 180 x = 180 - 42.5

x = 137.5

(137.5, 42.5)


The sum of two numbers is 70 and their difference is 24. Find the two numbers.

Using Elimination to Solve a Using Elimination to Solve a Word problem:Word problem:


x = first number

y = second number

Therefore, x + y = 70



x + y = 70

x – y = 24


x + y =70

x - y = 24 2x = 94

x = 47

47 + y = 70

y = 70 – 47

y = 23

(47, 23)

Now you Try to Solve These Now you Try to Solve These Problems Using Problems Using Elimination.Elimination.

Solve1. Find two numbers whose sum is

18 and whose difference is 22.

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

MATRIX:MATRIX: A rectangular A rectangular arrangement of arrangement of numbers in rows and numbers in rows and columns.columns.

The The ORDERORDER of a matrix of a matrix is the number of the is the number of the rows and columns.rows and columns.

The The ENTRIESENTRIES are the are the numbers in the matrix.numbers in the matrix.

502

126rows

columns

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

67237

89511

36402

3410

200

318 0759

20

11

6

0

7

9

3 x 3

3 x 5

2 x 2 4 x 1

1 x 4

(or square matrix)

(Also called a row matrix)

(or square matrix)

(Also called a column matrix)

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

34

03

12

70

43

35

)3(740

0433

13)2(5

44

40

23

9245

3108

2335

2571

)1(8 70 51 23

55 34 32 )2(9 =

= 7 7 4 5

0 7 5 7

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

232

451

704

831

605

429

2833)2(1

)4(65015

740249

603

1054

325

724

113

810

051

708

342

=

5-2

-4-1 3-8

8-3 0-(-1) -7-1

1-(-4)

2-0

0-7

=

2 -5 -5

5 1 -8

5 3 -7

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

14

024

416

08

)1(4)4(4

)0(4)2(4

86

54

30

212

)8(360

52412

-2

6

-3 3

-2(-3)

-5

-2(6) -2(-5)

-2(3) 6 -6

-12 10

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review 2.1-2.3. ex: check whether the ordered pairs are solns. of the system. x-3y= -5 -2x+3y=10...

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