math 8 - notes · web viewsolving linear equations unit lesson 1 – solving one-step equations...

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Solving Linear Equations Unit

Lesson 1 – Solving one-step equations

Learning Target #3 – I can solve one-step equations and show appropriate computation(s) performed

An equation is a mathematical sentence that includes two expressions that have the same ________________________.

Every equation has an _______________ sign and contains at least one _______________.

An equation looks like:

Finding a value for the variable which makes the equation a true statement is called ______________________________.

When an equation has been solved the answer will appear as:

An equation is like a balance beam. If something is done to one side of the balance beam then the same thing must be done to the _______________ side in order to maintain the _______________.

You can simplify equations algebraically by:

(a) ____________ or ____________ the same number from ____________ sides.

(b) ____________ or ____________ the same number from ____________ sides.

Two key things to remember when equation solving:

1. The variable is being ____________ so that it ends up being by itself on one side of the equation.

2. What is done to one side of the equation must be done to the __________________ to maintain balance.

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Here we go! Solve the following equations using one step.

1. n – 9 = 3 2. m + 3 = 0 3. x – 3 = -11

4. x + 12 = -25 5. -1 = y + 7 6. 8 + x = 2

7. x – 1 = -5 8. 4 + x = 16 9. 6n = 30

10. -21 = 7y 11. -5x = -35 12. 8y = -64

13. 16m = 12 14. -9x = 315.

16. 17. 18.

How do you determine whether to add, subtract, multiply, or divide to both sides of the equation?

Complete the following practice: What did the butcher….; Extra: textbook p. 377/378, #7-20

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Lesson 2 – Solving two-step equations

Learning Target #4 – I can solve basic two-step equations by showing appropriate steps and order

Steps to solving two-step equations:

1. Add or subtract the _____________ from both sides of the equation.

2. Divide both sides of the equation by the ____________________ORMultiply both sides of the equation by the __________________

OR multiply both sides by the reciprocal of the coefficient.

3. CHECK by substituting in the equation.

1. 2y + 4 = 12 2. 4y + 15 = 9 3. -5x + 12 = 10

4. -3n – 25 = 10 5. -4m – 5 = 15 6. -3y – 2 = -14

7. 5 + 7y = -51 8. -12 – 3x = -5 9. -17 = 2x - 5

10. 0 = -2x – 18 11. -5x = -3512.

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13. 14. 15.

16. 17. 18. -4.6n – 1.8 = 21.2

Complete “Vive La France!”; Connect Four game

Extra: textbook pages 385/386, 7-11; pg. 392, 6-12

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Lesson 3 – Collecting Like Terms and Evaluating Expressions

Learning Target #5 – I can simplify expressions by combining like terms.

Collecting or combining like terms is when you add or subtract like variables or constants

For example: 2x + 5x =

11t – t =

15p – 9p + 1 + 3 =

In many cases you will have more than one variable to combine

For example: 8a – 2b – 6a – 3b =

5c + d – 2c – d =

4xy – 5x + 3yx + 2y + 8y =

3x – 4 + 9c + 12 =

-3x2 – 8xy + 7x2 – 4xy =

Substitution is when you plug in the number into the given equationFor example: Simplify, then evaluate

Learning Target #6 – I can evaluate expressions by substituting values given for variables.

(a) 2a + 8a , when a = 2 (b) 7t – 3t, when t = 3

(c) 3 + 4x – 2x, when x = 5 (c) -5p – 3p + 1 – 6, when p = -1

Simplify and then evaluate each expression when x = -2 and y = 3.

(a) 3x + 4y – 5x – 2y b) 7x2 – 2y – 8y – 2x2

Complete practice sheets that follow: Combining Like Terms and Evaluating Expressions & “Silkworm” 119

Combining Like Terms PracticeSimplify the following by combining like terms

1. 2p – 5p 2. 5r + 2r + 16 -5 3. 15 + 9s – 2s

4. -4x + 7 + 2x 5. -2t + 9 + 5t -12 6. n -3 + 6n +5

7. x x 0 8. n n 9. -7mn + 2n + 5mn - 10n

10. 3x2 – 7x + 9 + 7x2 – 2 11. 11k -20k +5k +7 12. 4n -2m +5n –n +3m

Find the value of the expression if x = -3 and y = 2 (simplify before substituting values)

1. 5x – 3y 2. 16 -2x + 7x + 12

3. 7x2 – 4x2 - 27 4. –19y + 15x – 5x + 10 y – 18

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Warm-up Before Quiz 2

Simplify the following expressions. The evaluate for a = -1 and b = 2

1) 8ab + 3a2b – 4ab – 7ab2 + 2ab – 8a2b + 10ab2 + 4 – 15

2) -5 + 3a – 4b + 7 – 8a + 9b + a -2

Complete the table of values showing all steps. Then graph on the grid provided. Connect and label points.x y = -3x + 2 (x, y)

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x y = -3x + 2 (x, y)

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Lesson 4 – Collecting Like terms and Solving Equations

Example 1: 4x + 6 + 3x = 20

Example 2: -6m – 3 + 5m + 8 = -25

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Example 3: 5y + 3y – 2 = 62

Example 4: 5n + 2n – 9.1 = 20

Complete Practice that follows: What problem did the dumb gangster have when boss told him to blow up a car? & “What is the Title of this Picture?”

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Lesson 5 – Modelling with words – Translating English words to Mathematical Symbols

Learning Target #7 - I can translate words to mathematical expressions or equations.

Addition

Subtraction

Multiplication

Division

Equal

When translating start with the words “a ___________________”

Use a variable to represent the unknown number such as _____, _____, etc.

Examples

Four more than a number is 20.

Three times a number is -27.

A number is increased by seven is 12.

Four less than a number is six.

Twice a number increased by three is 33.

Three more than five times a number is 38.

One third of a number diminished by fifteen.

The sum of 10 and a number is negative 25.

The quotient of a number and 2 is thirteen.126

Writing Expressions Practice

Write an expression for each of the following using “n” as the variable or “m” if two variables needed.

a. Twice a number ____________________ b. Three more than a number________________

c. Ten less than a number__________________ d. A number increased by twenty____________

e. The product of six and a number_____________ f. A number divided by five________________

g. The quotient of 15 and a number_____________ h. Three more than triple a number __________

i. Twenty less than five times anumber ________________________________

j. Ten less than the product of five and a number______________________________

k. Four more than a number plus twice another number______________________

l. One quarter a number diminished by seven________________________

m. The sum of two consecutive numbers_____________________________

n. The difference of twice a number and eight______________________________

_______________________ __________________________

Lesson 6 – Modelling with words and solving equations

There are 4 steps when solving a problem.

Problem: Twice a number increased by eight is twelve. What is the number?

1. “Let” statement e.g. Let “n” be the unknown number

2. Write an algebraic equation e.g. 2n + 8 = 12

3. Solve the equation e.g. 2n + 8 = 12

4. State the solution to the problem e.g The number is _______.

Solve each of the following problems using the 4 steps.

1. Three more than the number is eight. What is the number?

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2. Three times a number is twenty-four. What is the number?

3. Twelve less than a number is seven. What is the number?

4. Three more than twice the number is twenty-five. What is the number?

5. Seven more than three times the number is -14. What is the number?

6. The product of six and a number then decreased by six is -30. What is the number?

7. A classroom’s length is 3 m less than two times its width. The classroom has a length of 9 m. What is the width of the classroom?

Complete: Worksheet- Writing Expressions and Solving Equations on the following page

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Writing and Solving Equations Practice

Write an equation for the following questions and then solve the equation. Substitute the answer back into the equation to check that the solution is correct.

How do equations differ from expressions?________________________________________________

a. Four more than a number is 25 b. Twice a number is negative thirty four

c. Five times a number is negative seventy five

d. Three less than a numbers is sixteen.

e. Six more than twice a number is 14 f. Twelve less than 5 times a number is 38.

g. A number plus, three more than the number is 33.

h. Twice a number diminished by six then increased by the number is thirty.

i. Twelve less than one third a number is 24.

j. Four times a number minus two times a number is 18.

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k. Three times a number plus five times a number is thirty two.

l. Seven more than two-fifths a number is 4.

m. Six times a number increased by 7 is 103. n. When 13 is subtracted from three-eighths of a number the result is 11.

o. Eight more than a number divided by 3 is 32.

p. Two consecutive numbers add up to 21. What is the smaller number?

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Lesson 7– Algebraic Problem Solving

Learning Target #8 – I can problem-solve by identifying the equation and showing all steps to achieve appropriate answer.

For each of the following problems complete:1. Write a “Let” statement – to define the variable(s)2. Write the equation3. Solve the equation using algebra4. Solution to the problem – make a statement

Problem 1: A shirt and pants cost $89. If the shirt cost $16 more than the pants, what is the cost of each?

Problem 2: Jill is three years younger than her brother Ken. The sum of their ages is 43. How old are Jill and Ken?

Problem 3: Mark scored twice as many baskets as Thomas. If their combined score was 45 baskets, how many did each score?

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Problem 4: For the month of January, the average afternoon temperature in Calgary is ¼ the average morning temperature. The average afternoon temperature is -4 °C. What is the average morning temperature?

(a) If m represents the average morning temperature, what equation models this problem?(b) Solve this equation. Verify your answer.

Problem 5: A cow sleeps 7 h a day. This amount of sleep is 1 h less than twice the amount of an elephant sleeps a day. How long does an elephant sleep?

Problem 6: The cost of a concert ticket for a student is $2 less than one half of the cost for an adult. The cost of the student ticket is $5. Let a represent the cost of an adult ticket. Write and solve an equation to determine the cost of an adult ticket.

Complete: Problem Solving –“Did you hear about….”

Review: Textbook pages 400-401 #1-4, 6, 9-10, 13-15, 17-19; p. 402 #1-4, 6, 10 (a-e), 12, 14

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Warm-Up Before Quiz 3

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