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Solving Systems of Linear Equations and Matrices

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Name:___________________________ Period:___________

College Bound Math Teacher:_______________

Reference Sheet

|Axis of symmetry: |[pic] |

|Quadratic Formula: |[pic] |

|Determinate of a 2×2 matrix |[pic] |

|Inverse of a 2×2 matrix |[pic]( [pic] |

|I = interest B = Balance P = principle |

|r = rate (as a decimal) n = number of compounding periods t = time in years |

|Exponential growth & decay |B = P(1 + r)t B = P(1 – r)t |

|Simple Interest |I=Prt |

|Compound Interest/Single Deposit Investment | [pic] [pic] |

|Future Value/Present Value | |

|Periodic Deposit Investment |[pic] [pic] |

|Future Value/Present Value | |

|Continuous Interest |B=Pert |

|Trigonometric Ratios |Sin A = [pic] Cos A = [pic] Tan A = [pic] |

|Coordinate Geometry |[pic] |

|Slope – Intercept Formula |y – y1 = m(x – x1) |

|Law of Sines |[pic] |

|Law of Cosines |[pic] [pic] |

|Area of a Triangle |[pic] |

Lesson 1: Solving Systems Graphically

Lesson Objectives:

The student will be able to determine if an ordered pair is part of a linear system

The student will be able to determine graphically the solution to a linear system

A system of linear equations is:_______________________________________________________

_________________________________________________________________________________

A solution to a system of linear equations is:_____________________________________________

_________________________________________________________________________________

Indentifying Solutions of Systems

Tell whether the ordered pair is a solution to the given system: PLUG IT IN!!!

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

x + 2y = 6 2x + 5y = 8 y = -2x + 5

x – y = 3 3x – 2y = 5 y = 2x + 1

Number of Solutions

Systems can have…

|One Solution |No Solutions |Infinitely Many Solutions |

|(Intersect once) |(Never Intersect) = Parallel Lines | |

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Solve Systems by Graphing – where do the lines intersect?

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Homework #1: Solving Systems Graphically

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5) 3x – 2y = 8 6) 2x + y = 5

4y = 6x – 4 10x + 5y = 25

Lesson 2: Solving Systems of Equations by Substitution

The student will be able to solve a system of equations by substitution

Solving by substitution is an arithmetic method.

Step 1 Solve for one __________________________ in at least one equation if necessary.

Step 2 ________________________ the resulting expression into the other equation.

Step 3 Solve the equation to get the value of the first variable.

Step 4 Substitute that value into one of the ___________________ equations and then solve.

Step 5 Write your values from Step 3 and 4 as an ______________________

Examples:

y = 3x – 1 y = 3x – 5

2x + y = 9 y = x + 3

3x – y = 7 2x + 4y = 8

2y = 6x + 5 3x + 5y = 14

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Lesson3: Solving Systems of Equations by Elimination

Student will be able to Solve a System of Equation by Elimination

Solving by substitution is an arithmetic method.

Step 1 Write the system into standard form _________________________________

Step 2 Determine the variable to eliminate by _______________________________

____________________________________________________________________

Step 3 Eliminate for one variable by using the method determined in Step 2

Step 4 Substitute the value into one of the ________________ original equations and then solve.

Step 5 Write the values from Steps 3 and 4 as an ____________________________

Examples:

x – 2y = 5 3x + y = 14 2x + y = 5

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

x + 2y = 6 3x + 4y = -25 6x + 4y = 42

3x + 3y = -6 2x – 3y = 6 8x – 7y = 19

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Lesson 5: Mixed Elimination & Substitution

Solve the following system of equations using any method.

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Name:___________________________________ Date:______________________

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Lesson 6: Intro to Matrices, Addition, Subtraction & Scalar Multiplication

Students will learn about matrices

Students will know how to perform addition & subtraction on a matrix

Students will know how to perform scalar multiplication on a matrix

Matrix (matrices) – a rectangular array of numbers. Named by their Rows х Columns.

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The entries or elements are named by its position in the matrix.

If two matrices have the same dimensions, you can add or subtract them. To do this you must add or subtract elements in the same position. If the two matrices are NOT the same dimensions, you CANNOT add or subtract them.

Add the following matrices.

1. 5 8 -2 3 -2 4 2. -1 0 5 -2

0 4 -1 + -4 2 3 2 1 + -1 1

3 1 0 7 -8 4 4 -3 -3 2

Subtract the following matrices.

3. 5 8 -2 3 -2 4 4. -1 0 5 -2

0 4 -1 - -4 2 3 2 1 - -1 1

3 1 0 7 -8 4 4 -3 -3 2

Can not be added or subtracted, why?

5.

Scalar multiplication – multiplying the matrix by a constant. – “distribute” the constant to EACH element inside the matrix.

Multiply the following matrices.

6. 5 8 -2 7. -1 0

-2 0 4 -1 [pic] 2 1

3 1 0 4 -3

Simplify the following expression.

8. 5 8 -2 3 -2 4 9. -1 0 5 -2

-3 0 4 -1 + 5 -4 2 3 2 2 1 - 2 -1 1

3 1 0 7 -8 4 4 -3 -3 2

Homework #6: Intro to Matrices, addition, subtraction & Scalar Multiplication

Lesson 7: Matrix Multiplication

The students will be able to multiply two matrices of at most 3x3

To multiply a matrix by another matrix, they do NOT have to be the SAME dimension, but the number of ROWS of one matrix MUST BE the same as the number of COLUMNS of the other (Match up the middle numbers). Your resulting matrix will be what’s on the outside. (See picture below).

For example:

|2 x 2, 2 x 2 |3 x 2, 2 x 4 |7 x 1, 1 x 3 |2 x 5, 6 x 2 |1 x 4, 2 x 5 |

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So how do we get the numbers that go INSIDE the matrix? We use the DOT product (Multiply AND add).

[pic] and [pic]

| | | |6 |7 |5 |

| | | |2 |1 |0 |

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|0 |2 | |0∙6 + 2∙2 |0∙7 + 2∙1 |0∙5 + 2∙0 |

|1 |4 | |1∙6 + 4∙2 |1∙7 + 4∙1 | |

|5 |1 | |5∙6 + 1∙2 | | |

|3 |1 | | | | |

Find the dimensions of each matrix product.

Ex. 2 A5X2 • B2X5 Ex. 3 M4X2 • N2X3 Ex. 4 C2X3 • B2X3 Ex. 5 Q1X4 • R4X2

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Homework #7: Matrix Multiplication

SHOW ALL WORK!!!

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Lesson 8: Determinates

The students will be able to calculate the determinate of a 2х2 matrix.

Determinants Date: _______________________________

A determinant of a matrix represents a single number. We obtain this value by multiplying and adding its elements in a special way. We can use the determinant of a matrix to solve a system of simultaneous equations.

Properties of Determinants

• The determinant is a real number, it is not a matrix.

• The determinant can be a negative number.

• It is not associated with absolute value at all except that they both use vertical lines.

• The determinant only exists for square matrices (2×2, 3×3, ... n×n). The determinant of a 1×1 matrix is that single value in the determinant.

• The inverse of a matrix will exist only if the determinant is not zero. If the determinant is zero your system has NO INVERSE! For a system there will be NO SOLUTION or INFINITLY MANY SOLUTIONS.

Why do we care about determinants?

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You find the determinant using the formula: [pic]

Examples:

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|Example 1: [pic] |Example 2: [pic] |

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|Example 3: [pic] |Example 4: [pic] |

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|Example 5: [pic] |Example 6: [pic] |

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Homework #8: Determinates

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Lesson 9: The Inverse Matrix

The students will be able to determine if an inverse exists.

The students will be able to calculate the inverse of a 2х2 matrix

Inverse of a Matrix

Why care about the inverse of a matrix?

▪ The inverse of a matrix can help us solve a system of equations without graphing, or using substitution or elimination.

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A Matrix has an inverse (solution) IF AND ONLY IF its determinant is NOT ______________!!!

Practice:

For each matrix state if an inverse exists.

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Examples:

Find the inverse of each matrix

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Practice: Find the inverse of each matrix, if it exists.

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5) [pic] 6) [pic] 7) [pic] 8) [pic]

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Lesson 10: Using Matrices to Solve Equations

Students will be able to solve a system of equations by using a matrix

When we solve an equation that has only one variable, we are finding the value for that variable that makes the equation true. If our equation has two variables, there can be infinitely many combinations of numbers that would work. For example, if we have an equation like [pic], values of x and y could be -1 and -2, -3 and 2, or any other combination of numbers.

A system of equations is when we have more than one equation and more than one variable. For example:

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We refer to this as a system of equations, meaning that we want x and y values that make BOTH equations true.

|Steps to Solving Systems using Matrices |Example: [pic] |

|1.) Set up your Matrix Equation. | |

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|Create your “Coefficient Matrix” | |

|It is called the coefficient matrix because it is created by using the | |

|coefficients of the variables involved. | |

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|Create your “Constant Matrix” | |

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|It is created from the constants on the right side of the equal signs. | |

|2.) To solve ANY equation (get the variables alone), we use the INVERSE | |

|(OPPOSITE). So now we must find the inverse of our Coefficient Matrix to get rid | |

|of it! | |

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|Multiply both sides of the Matrix equation by the inverse to get your variables | |

|alone. | |

|6.) You have your answer!!! | |

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Examples:

Solve the systems of equations using matrices

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Homework #11: Solving Systems of Equations with Matrices

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Lesson 11: Solving Systems using any Method

Students will be able to solve a system of equations using any chosen algebraic method (substitution, elimination, matrix)

Practice: If [pic], then A-1 = [pic], where |A| = ad – bc

Solve the following systems of equations using a matrix

1) 11x + 5y = 5 2) -11x – 6y = -22

10x – 6y = -6 3x + 8y = 6

Solve the following systems of equations using either substitution, elimination or using a matrix.

3) x – 2y = 5 4) 2x + 2y = 4

3x – 5y = 8 x – 2y = 5

5) 2x + 3y = 11 6) y = -6x + 17

5x – 2y = -20 4x + 9y = 3

7) y = 10x – 13 8) 4x + 7y = -17

7x + 3y = -2 9x + 8y = 16

9) y = 4x – 8 10) x + 4y = -20

y = 6x – 12 x – 5y = 16

11) 6x – 2y = -20 12) 6x + 3y = -15

6x + 6y = 12 3x + 2y = -11

13) y = 3x + 28 14) 4x + 8y = -16

2x = 13 – y 6x + 12y = -24

15) x = 7y – 8 16) 3x + 3y = -24

4x + 8y = 4 -2x + 2y = 24

Lesson 12: System of Equations Word Problems

The students will be able to solve system of equation word problems.

1) Find the value of two numbers if their sum is 12 and their difference is 4.

2) The difference of two numbers is 3. Their sum is 13. Find the numbers.

3) Kristin spent $131 on shirts. Fancy shirts cost $28 and plain shirts cost $15. If she bought a total of 7 then how many of each kind did she buy?

4) Chase and Sara went to the candy store. Chase bought 5 pieces of fudge and 3 pieces of bubble gum for a total of $5.70. Sara bought 2 pieces of fudge and 10 pieces of bubble gum for a total of $3.60. Which system of equations could be used to determine the cost of 1 piece of fudge, f, and 1 piece of bubble gum, g?

5) At an ice cream parlor, ice cream cones cost $1.10 and sundaes cost $2.35. One day, the receipts for a total of 172 cones and sundaes were $294.20. How many cones were sold?

6) You purchase 8 gal of paint and 3 brushes for $152.50.The next day, you purchase 6 gal of paint and 2 brushes for $113.00. How much does each gallon of paint and each brush cost?

7) Shopping at Savers Mart, Lisa buys her children four shirts and three pairs of pants for $85.50. She returns the next day and buys three shirts and five pairs of pants for $115.00.What is the price of each shirt and each pair of pants?

8) Grandma’s Bakery sells single‐crust apple pies for $6.99 and double‐crust cherry pies for $10.99.The total number of pies sold on a busy Friday was 36. If the amount collected for all the pies that day was $331.64, how many of each type were sold?

9) Suppose you are starting an office‐cleaning service. You have spent $315 on equipment. To clean an office, you use $4 worth of supplies. You charge $25 per office. How many offices must you clean to break even?

10) The math club and the science club had fundraisers to buy supplies for a hospice. The math club spent $135 buying six cases of juice and one case of bottled water. The science club spent $110 buying four cases of juice and two cases of bottled water. How much did a case of juice cost? How much did a case of bottled water cost?

11) At The Apple Pan, 4 burgers and 3 fries cost $26.50. 5 burgers and 5 fries cost $36.25. What is the cost for each item?

12) An army of goblins and orcs eats breakfast before a busy day of burning and pillaging. Each orc eats 4 pieces maggot bread while each goblin eats 3 pieces of maggot bread. The army eats 265 pieces of maggot bread. Each orc has 3 weapons. Each goblin has 2 weapons. The army has 190 weapons.

Lesson 13: More System of Equation Word Problems

The students will be able to solve system of equation word problems.

1) The senior class at Northwest High School needed to raise money for the yearbook. A local sporting goods store donated hats and T-shirts. The number of T-shirts was three times the number of hats. The seniors charged $5 for each hat and $8 for each T-shirt. If the seniors sold everything and raised $435, what was the total number of hats and the total number of T-shirts that were sold?

2) The sophomore class at South High School raised $800 from the sale of tickets to a dance. Tickets sold for $1.50 in advance and $2.00 at the door. If a total of 475 tickets were sold, what was the number of tickets sold at the door?

3) A craft shop sold 150 pillows. Small pillows were $6.50 each and large pillows were $9.00 each. If the total amount collected from the sale of these items was $1180.00, what is the total number of each size pillow that was sold?

4) Three bags of potatoes and four cases of corn cost $41.35. Five bags of potatoes and two cases of corn cost $36.95. Find the cost of one bag of potatoes and the cost of case of corn.

5) A museum sold a total of 230 adult and children’s tickets. If each adult ticket cost $7.50 and each children’s ticket cost $5.75. If the museum had total receipts of $1497.50, how many of each type ticket did they sell?

6) Brenda's school is selling tickets to a spring musical. On the first day of ticket sales the school sold 30 senior citizen tickets and 75 child tickets for a total of $450. The school took in $600 on the second day by selling 48 senior citizen tickets and 90 child tickets. What is the price each of one senior citizen ticket and one child ticket?

7) Eldora and Finn went to an office supply store together. Eldora bought 15 boxes of paper clips and 7 packages of index cards for a total cost of $55.40. Finn bought 12 boxes of paper clips and 10 packages of index cards for a total cost of $61.70. Find the cost of one box of paper clips and the cost of one package of index cards.

8) Ilida went to Minewaska Sate Park one day this summer. All of the people at the park were either hiking or bike riding. There were 178 more hikers than bike riders. If there were a total of 676 people at the park, how many were hiking and how many were riding their bikes?

9) A catering company is setting up tables for a big event that will host 764 people. When they set up the tables they need 2 forks for each child and 5 forks for each adult. If the company ordered a total of 2992 forks, how many adults and how many children will be attending the event?

10) Five yards of fabric and three spools of thread cost $40.12. Two yards of the same fabric and ten spools of the same thread cost $23.88. Find the cost of a yard of fabric and the cost of a spool of thread.

11) Half a watermelon and a half pound of cherries cost $3.09. A whole watermelon and two pounds of cherries cost $8.16.

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Resulting Matrix:

Resulting Matrix:

Resulting Matrix:

Resulting Matrix:

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