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Algebra 1 – Spencer – Unit 3 Notes Chapter 3 Solving Linear Equations

Unit Calendar

Date Sections Topic Homework HW √ October 7 (A) October 8 (B)

3.1 3.2 3.3 3.4

One-Step Equations Two-Step Equations Multi-Step Equations Equations with variables on both sides

Solving 5 Different Types of Equations Packet

October 9 (A) October 13 (B)

3.8 Quiz on 3.1 – 3.4 Literal Equations

Pg 187: 3-17 odd, 33

October 14 (A) PSAT October 15 (B) October 16 (A)

3.5 3.6

Writing Ratios/Proportions Solving Proportions

Pg 165: 1-10, 21, 25, 27, 2 Pg 171: 4, 8, 12, 18, 20, 24, 28

October 19 (B) October 20 (A)

3.7 Quiz on 3.5, 3.6, 3.8 Solving Percent Problems

Pg 179: 3 – 19 odd, 26, 29, 35


Chapter 3

Review Day Whatever isn’t completed in class


Chapter 1-2

Quarter 1 Review Stations Whatever isn’t completed in class


Chapters 1-3

Quarter 1 Test


Quarter 1

Quarter 1 Project/Activity

SOL A.4: The student will solve multistep linear equations in two variables, including:

SOL A.4a: Solve literal equations (formulas) for linear situations (2 variables); SOL A.4b: Justify steps used in simplifying expressions and solving equations, using properties and axioms of equality that are valid for the set of real numbers and its subsets; SOL A.4d: Solve multistep linear equations algebraically SOL A.4f: Solve real-world problems involving equations

Vocabulary: inverse operations, equivalent equations, reciprocal, like terms, input, output, distributive property, identity, ratio, proportion, simplest form, cross product, scale drawing, scale model, scale, percent, proportion. Textbook resources: Chapter 3

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Solving Equations – One-step Equations The goal of solving an equation is to isolate the variable. Use inverse operations to undo what is being done to the variable. Inverse operations are operations that are opposites of one another. Start by undoing the easiest operations first! Think reverse order of operations. Which operation is the inverse operation of:

Addition Multiplication Division Subtraction

We use the properties of equality to isolate the variable.

The equation must always stay equal. Whatever you do to one side you must do the other on the same step! The Golden Rule of Algebra is “Do unto one side what you do to the other.” *If it is on the same side of the equal sign COMBINE LIKE TERMS!*

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Check ex 1: x + 3 = 10 + 5

x + 3 = 15 - 3 -3 x = 12 ex 2: n – 5 = -8 + 6 n – 5 = -2 + 5 + 5 n = 3 ex 3: 3x = |-15 |

ex 4: 67x

=

ex 5: ( )( )3 6 44

n = −

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White Board Practice: Solve. Check Check A) 3 14y − = − B) -7 = x + |4|

C) 92y

− =−

D) 48 6a= −

E) 2 303

x = −

Solving Two-Step Equations A two-step equation involves using two-steps to solve for the variable.

To solve two-step equations: 1) Add or subtract away whatever is “bothering” the

variable 2) Multiply or divide to isolate the variable 3) Check by substitution

Example: A) 2 1 9x + = What is being done to the variable? What is the plan to undo it? Solve and check. Check Check

B) 3 4 15 2x − = + C) 33 25n

= +

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Check Check

D) 5 3 2y+ = E) 8 57

n= −−

Sometimes the coefficient is a fraction, so you can multiply by the reciprocal AFTER you have done any necessary addition or subtraction. Reciprocal: ____________________________________________

_____________________________________________________

Check Check Ex: ⅔X − 9 = 15 10 = ⅛n + 4 REMINDER: -x has a coefficient of -1, because –x = -1x. If you reach a step in your equation that has –x = (any number), then multiply or divide both sides by -1. Ex: 4 – x = 6 -4 -4 (subtract 4 from both sides) -x = 2 -1(-x) = 2(-1) (multiply both sides by -1) x = -2 To check: 4 – (-2) = 6 ? 4 + (+2) = 6 6 = 6 YES!

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Try these: 1) 12 – n = 5 4) -38 = ¾x - 2 2) ⅝n + 3 = 23 5) 52 = 60 – x

3) 10 304x−+ = 6) 7 8

2n

− − =

Multi-Step Equations The first step when solving equations is to simplify each side of the equal sign. For some equations you need to combine like terms. Ex : a) 5x + 2x = 28 b) -9x – 2x = 66 7x = 28

7 287 7x =

x = 4

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For other equations you need to use the distributive property first. Ex: e) 6(2x + 3) = 42 f) -2(5n – 1) = 7 12x + 18 = 42 -18 -18 12x = 24

12 2412 12

x =

x = 2 g) 4x + 3(8x – 5) = -15 h) 7(n – 3) + 1 = 62 4x + 24x + (-15) = -15 28x + (-15) = -15 + 15 + 15

28x = 0 28 28 x = 0 Try these: 1) 6x + 3x = 36 2) 4(2x + 1) = 20 3) 2(5x – 3) = 34 4) -4x – 8 + 3x = 17 5) 5(4n + 3) + 10 = 105 6) 3(3n -7) - n = 19

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Solving Equations with Variables on Both Sides Some equations have variables on both sides. To solve, you want to move the variable terms to one side and the constant terms to the other side of the equation. Then, isolate the variable. Steps:

1) Simplify each side of the equation, if necessary. 2) Get all variables on one side of the equal sign by adding or

subtracting. 3) Solve as you would any multi-step equation.

1) 5n – 4 = 3n + 6 2) 2y = 27 - y 3) 6x + 14 + 2x = 35 + x 4) 4(2x + 5) + -6² = x + x + 24 5) ⅓(12n - 9) + 1 = 2n + 3(n – 5) 6) ½(12 – 2x ) – 4 = 5x + 2(x – 7)

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Number of Solutions Equations do not always have one solution. An equation that is true for all values of the variable is called an identity. You have an identity when the result of solving an equation is always true. The solution of an identity is all real numbers. What does that mean? Example Practice 3(𝑥𝑥 + 4) = 3𝑥𝑥 + 12 2𝑥𝑥 + 10 = 2(𝑥𝑥 + 5) 3(2𝑎𝑎 + 2) = 2(3𝑎𝑎 + 3) When the result of solving the equation is never true, then there is no real solution to the equation. Example Practice 9𝑧𝑧 + 12 = 9(𝑧𝑧 + 3) 3(4𝑥𝑥 + 2) = 2𝑥𝑥 + 2(5𝑥𝑥 + 1) Real-World Problem Practice

1. The sum of two consecutive numbers is 43. What are the numbers?

2. The length of a rectangle is three times the width. If the length is 24 inches, what is the width?

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3. Jose is 2 more than twice Mary’s age. If Jose is 20, how old is Mary?

4. Non-members pay $25 to play a round of golf at the local course. Members pay a fee of $100 plus $10 for each round. For how many rounds of golf is the total cost the same for non-members and members to play golf?

Rewrite Equations and Formulas – Equations with Multiple Variables

A literal equation is an equation that involves two or more variables.

Example: You are ordering pizzas and sandwiches. You have a budget of $80. Pizzas cost $10 each. Sandwiches are $5 each. How many sandwiches can you buy if you buy 4 pizzas? 1) Create the literal equation:

10p + 5s = 80, where x is the number of pizzas and y is the number of sandwiches that you can buy.

Why 10p? Why 5s? Why 80?

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2) We want to know how many sandwiches we can buy, so we want to solve the equation for y:

10x + 5y = 80 -10x -10x 5y = 80 – 10x 5 5 y = 16 – 2x

3) Substitute 4 into the new equation for x and solve.

Y = 16 – 2(4) Y = 16 – 8 Y = 8 Therefore, if you buy 4 pizzas, you can buy 8 sandwiches with your $80 budget.

Try this one: 4 = 2m – 5n What is the value of m when n = 3? What is the value of m when n = -4?

Subtract 10x from both sides to get “y” alone on one side of the equation. Divide both sides by 5 to isolate the variable – y. Simplify: 80 ÷ 5 = 16; -10x ÷ 5 = -2x

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Literal equations may only have variables. For example: -t = r + px Let’s solve for x. -r -r -t – r = px p p (-t – r) = x p

Try these on your own: Solve for y: ax + by = c Solve for b: A = 1/2h(b + c) Solve for R: P = R - C N

Common Literal Equations with which you should be familiar:

A = L * W D = r * t

Subtract r from both sides. Divide both sides by p to isolate x.

Solved!

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More Practice

Solve for the indicated variable in the parenthesis.

1) P = IRT (T) 2) A = 2(L + W) (W)

3) y = 5x - 6 (x) 4) 2x - 3y = 8 (y)

5) x + y = 5 (x) 6) y = mx + b (b) 3

7) ax + by = c (y) 8) A = 1/2h(b + c) (b)

9) V = LWH (L) 10) A = 4πr2 (r2)

11) V = πr2h (h) 12) 7x - y = 14 (x)

13) A = x + y (y) 14) R = E (L) 2L

15) x = yz (z) 16) A = r (L) 6 2L

17) A = a + b + c (b) 18) 12x – 4y = 20 (y) 3

19) x = 2y - z (z) 20) P = R - C (R) 4 N

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Chapter 3.5 and 3.6 Writing and Solving Ratios and Proportions

Ratio: a relationship between two quantities that can be written in the following ways:

:

is to

a baba b

*Ratios should be written in simplest form.*

Rate: any comparison of a per b format . . mii ehr

Unit Rate: any comparison of a per 1 unit 60sec. .1min

i e

Percentage: literally means “out of 100”

PART %

WHOLE 100

Write a ratio for the following: a) 300 miles to 15 gallons

b) 10 cups to 2 ounces

c) 3 pears to 6 watermelons

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Write unit rates for the following:

a) 300 miles to 15 gallons

b) 10 cups to 2 ounces

c) 3 pears to 6 watermelons

Proportions are two ratios that are equivalent to one another. They are generally used to find similar measures.

They are written in one of the three following methods.

: :

is to as is to .

a b c da cb da b c d

=

=

In each example above, a and d are known as the “extremes,” and b and c are known as the “means.” You solve proportions by “cross-multiplying” or multiplying the means and the extremes and creating an equation. Such as:

a d b c• = •

Then you simply solve the resulting equation.

YOU CAN ONLY CROSS MULTIPLY WHEN YOU HAVE TWO EQUAL PROPORTIONS. YOU CANNOT CROSS MULTIPLY WHEN ADDING OR SUBTRACING FRACTIONS.

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Examples: Write the ratio in simplest form.

1) 1814

2) 5 is to 13

Solve the proportion.

3) 35

2 x=

4) 212

36 x=

5) 6440

83=

+m

6) 39

4135 −=

x

Write the sentence as a proportion. Then solve the proportion.

7) 3 is to 8 as x is to 32.

8) y is to 20 as 9 is to 5.

9) t is to 21 as 40 is to 28.

10) 4 is to 12 as n is to 3.

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11) A 5 foot tall student stands near a flagpole. The flagpole and the student are perpendicular to the ground. The sun’s rays strike the flagpole and the student at the same angle, forming 2 similar triangles. The flagpole casts a 15 foot shadow, and the student casts a 2 foot shadow. Use indirect measurement to find the height of the flagpole. (Hint: use a proportion)

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Chapter 3.7 SOLVING PERCENT PROBLEMS

*is means equal* *of means multiply*

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Problem Solving Tip:

IS = % Example 1 above: 51136

= 𝑥𝑥%100

. Solve the proportion! OF 100 Extra Practice What percent of 96 is 12? What number is 35% of 18? 14 is 40% of what number? What percent of 125 is 30? What number is 250% of 18? What percent of 58 is 8.7? 30.1 is 35% of what number? What number is 70% of 250?

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