alg 1 notes 2 1 to 2 5 - ms. zaleski's math classes -...

Algebra I Name: of 13

CLASS NOTES: §2 – 1 thru §2 – 5 Solving Equations §2 – 1: Solving One-Step Equations Vocabulary:

• Equivalent equations – equations that have the same solution

• Isolate – get by itself

• Inverse operation – operations that undo each other. Add and subtract are inverse operations. Multiply and divide are inverse operations.

EX 1 Use the subtraction property of equality to solve each equation. (a) x + 13 = 27 (b) y + 2 = −6

Addition Property of Equality – Adding the same number to each side of an equation produces an equivalent equation. Algebra Example

If , then

Subtraction Property of Equality – Subtracting the same number from each side of an equation produces an equivalent equation.

Algebra Example

If , then


EX 2 Use the addition property of equality to solve each equation. (a) −7 = b − 3 (b) m − 8 = −14

(c) 12

= y −32

Multiplication Property of Equality – Multiplying each side of an equation by the same number produces an equivalent equation. Algebra Example

If , then

Division Property of Equality – Dividing each side of an equation by the same number produces an equivalent equation.

Algebra Example

If , then


EX 3 Use the division property of equality to solve each equation. (a) 4x = 6.4 (b) 10 = 15x (c) −3.2z = 14 EX 4 Use the multiplication property of equality to solve each equation.

(a) x4

= −9 (b) 19 =r3

(c) −x9

= 8

EX 5 Use reciprocals to solve each equation.

(a) 45m = 28 (b) 12 =

34x


§2 – 2: Solving Two-Step Equations

EX 1 Solve each equation using two steps. (a) 2x + 3 = 15 (b) 5x + 12 = −13

(c) 5 =t2

− 3 (d) 6 =m7

− 3

(c) x − 73

= −12

Steps for Solving a Two-Step Equation

• First, add or subtract on both sides of the equation.

• Second, multiply or divide on both sides of the equation.

What operation should you perform first? Multiplication. When you multiply by the denominator of the fraction in the equation, you get a new one-step equation with no fraction.


EX 2 Solve each equation using two steps.

(a) 6 =x − 24

EX 3 Solve each equation using two steps. State the rule that justifies each step. (a) −t + 8 = 3

(b) x3

− 5 = 4


EX 4 (a) You are making a bulletin board to advertise community service

opportunities for students. You plan to use half a sheet of construction paper for each ad. You need five sheets of construction paper for a title banner. You have 18 sheets of construction paper. How many ads can you make?

(b) Suppose you used one quarter of a sheet of paper for each ad and four

full sheets for the title banner in the last problem. How many ads could you make.


§2 – 3: Solving Multi-Step Equations EX 1 Solve each multi-step equation. (a) 5 = 5m − 23 + 2m (b) 11m − 8 − 6m = 22 (c) −2y + 5 + 5y = 14 EX 2 Noah and Kate are shopping for new guitar strings in a music store. Noah buys

2 packs of strings. Kate buys 2 packs of strings and a music book. The book costs $16. Their total cost is $72. How much is one pack of strings?

How is this equation different from equations you’ve seen before? The variable occurs in two terms. You can simplify the equation by grouping like terms and combining them.


EX 3 Solve each multi-step equation. (a) −8 2x − 1( ) = 36

(b) 18 = 3 2x − 6( ) EX 4 Solve each multi-step equation with fractions.

(a) 3x4

−x3

= 10

How can you make the equation easier to solve? Remove the grouping symbol by using the Distributive Property.

How do you get started? Clear the fractions from the equation by multiplying by the number that is the “common denominator” of all the fractions in the equation.

• Multiply both sides by 12 • Distribute the 12 on the left • Now multiply • Combine like terms • Divide


EX 5 Solve each multi-step equation with fractions.

(a) 2b5

+3b4

= 3 (b) 19

=56

−m3

EX 6 Solve the equation that contains decimals. (a) 3.5 − 0.02x = 1.24 EX 7 Martha takes her niece and nephew to a concert. She buys T-shirts and bumper

stickers for them. The bumper stickers cost $1 each. Martha’s niece wants 1 shirt and 4 bumper stickers, and her nephew wants 2 shirts but no bumper stickers. If Martha’s total is $67, what is the cost of one shirt?


§2 – 4: Solving Equations With Variables on Both Sides EX 1 Solve equations with variables on both sides. (a) 5x + 2 = 2x + 14 (b) 7k + 2 = 4k − 10 (c) 2 5x − 1( ) = 3 x + 11( ) (d) 4 2y + 1( ) = 2 y − 13( )

How do you get started? Add or subtract to get all the variables on one side only.

How do you get started? First, remove the parentheses by using the distributive property. Then add or subtract to move all variables to one side.


EX 2 What is the solution of each equation? (a) 10x + 12 = 2 5x + 6( ) (b) 9m − 4 = −3m + 5 + 12m (c) 3 4b − 2( ) = −6 + 12b (d) 2x + 7 = −1 3 − 2x( )

How can you tell how many solutions an equation has? If you eliminate the variable in the process of solving, the equation is either an “identity” with infinitely many solutions or an equation with no solution. Identity: an equation that is exactly the same on both sides.

Steps for Solving Equations

1. Multiply by the common denominator to remove fractions. Use the Distributive Property to remove parentheses.

2. Combine like terms on each side.

3. Add or subtract to get all of the variables on one side.

4. Add, subtract, multiply or divide on both sides to solve for the variable.

5. Check your answer by plugging it back in to the original equation.


§2 – 5: Literal Equations and Formulas Vocabulary:

• Literal equation – an equation that has two or more variables.

• Formula – an equation that states a relationship among quantities.

EX 1 Rewrite the literal equation. (a) 10x + 5y = 80; for y What is y when x = 3,6 ? (b) 4 = 2m − 5n; for m What is m when n = −2,0,2 ?


EX 2 Rewrite the literal equation. (a) ax + by = c ; for x (b) −t = r + px ; for x EX 3 Rewrite the formula. (a) What is the radius of a circle with circumference 64 ft? Round to the

nearest tenth. Use 3.14 for π. (b) What is the height of a triangle that has an area of 24 in2 and a base with

a length of 8 in? (c) The monarch butterfly is the only butterfly that migrates annually north

and south. The distance that a particular group of monarch butterflies travels is 1,700 miles one way. It takes a typical butterfly about 120 days to travel one way. What is the average rate at which a butterfly travels in miles per day? Round to the nearest mile per day.

alg 1 notes 2 1 to 2 5 - ms. zaleski's math classes -...

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