alg1 guided notes - unit 2 - solving equations

Unit Essential Questions

Can equations that appear to be different be equivalent?

How can you solve equations?

What kinds of relationships can proportions represent?

Williams Math Lessons

Algebra 1 Solving Equations -28-

WARM UP Tell whether the ordered pair is a solution of each equation. 1) y = 3x + 5; (1,8) 2) y = −2(x + 3); (−6,0)

KEY CONCEPTS AND VOCABULARY

_____________________________________________– have the same solutions and are the result of balancing an equation (whatever is done to one side of the equal sign has to be done to the other side).

_______________________ a variable means to have a variable with a coefficient of 1 by itself on one side of the equal sign.

___________________________________________________ – are operations which ‘undo’ each other. Subtraction and addition undo each other as well as multiplication and division.

_____________________________________________________________________ – adding /subtracting the same number to/from each side of an equation produces an equivalent equation.

_________________________________________________________________________________ – multiplying/dividing each side of an equation by the same number produces an equivalent equation.

EXAMPLES

EXAMPLE 1: SOLVING EQUATIONS BY ADDITION OR SUBTRACTION

Solve the following equations. a) b) −4 + r = 7

c) d)

−12 = 3+w

− 5

6+ j = − 1

6 t − 2

5= 1

3

ONE-STEP EQUATIONS MACC.912.A-CED.A.1: Create equations and inequalities in one variable and use them to solve problems.

MACC.912.A-REI.A.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution

method.

RATING LEARNING SCALE

4 I am able to

• use one-step equations to solve real-world applications or more challenging problems that I have never previously attempted

3 I am able to • solve one-step equations

2 I am able to

• solve one-step equations with help

1 I am able to • understand that I can solve problems by creating equations

TARGET


EXAMPLE 2: SOLVING EQUATIONS BY MULTIPLICATION OR DIVISION

Solve the following equations.

a) b) − x

3= −12

c) d)

EXAMPLE 3: SOLVING EQUATIONS WITH FRACTIONS

Solve the following equations.

a) b)

c) d)

EXAMPLE 4: SOLVING EQUATIONS FOR A REAL WORLD SITUATION

Shannon and Kelly spent $135 on gift certificates for their friends during the holidays. If this amount is 5/7 of their total spending money, how much spending money did they originally have? Set up an equation and solve.

RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)

Circle one: 4 3 2 1

5k = 45

−16g = 4

c−7

= 11

45

p = 153

−2y7

= 213

211

= −13

k −1

45

k = 267


WARM UP

Simplify each expression.

1) 4[3 – (3 – 2) – 2] 2) –1 + 2 × 4 – 6


Steps for Solving Two-Step Equations (ax + b = c )

• Add or subtract (undo) the ‘b’ value (to both sides) • Multiply or divide (undo) the ‘a’ value (to both sides) • Check answer

Steps for Solving Two-Step Equations (

x + ab

= c )

• Multiply (undo) the ‘b’ value (to both sides) • Add or subtract (undo) the ‘a’ value (to both sides) • Check answer

EXAMPLES

EXAMPLE 1: SOLVING TWO-STEP EQUATIONS: (ax + b = c)

Solve. a) 5q – 13 = 37 b) 2x + 3 = 18

c) 12a – 5 = 139 d) 7 + 5p = 52

TWO-STEP EQUATIONS MACC.912.A-CED.A.1: Create equations and inequalities in one variable and use them to solve problems.


method.


4 I am able to

• use two-steps equations to solve real-world applications or more challenging problems that I have never previously attempted

3 I am able to • solve two-step equations in one variable

2 I am able to

• solve two-step equations in one variable with help

1 I am able to • undo operations by working in reverse order from the order of operations

TARGET


e) f)

EXAMPLE 2: SOLVING TWO-STEP EQUATIONS: (

x + ab

= c )

Solve.

a) b)

EXAMPLE 3: WRITING AND SOLVING TWO-STEP EQUATIONS

Write an equation and solve. Eight more than five times a number is negative 62.

EXAMPLE 4: SOLVING EQUATIONS FOR REAL WORLD SITUATIONS

A calling plan charges $0.10 per minute and a monthly fee of $5.99. How many minutes can a customer talk if they want the bill to equal $37.99?


Circle one: 4 3 2 1

t12

− 9 = −11 0.2n + 3 = 8.6

x + 8−3

= −2

h − 35

= 10


WARM UP

Simplify each expression. 1) 8(2 – x) 2) –3(–x – 4)

3) 4(–2x + 1) 4) 6(3 – 2x)


Steps for Solving Multi-Step Equations • Simplify each side of the equation if possible (distribute, combine like terms) • Add or subtract (undo) the ‘b’ value (to both sides) • Multiply or divide (undo) the ‘a’ value (to both sides) • Isolate variable • Check answer

EXAMPLES

EXAMPLE 1: SOLVING MULTI-STEP EQUATIONS

Solve. a) 5+ 7t − 3− 4t = 9 b) 7p + 8p − 16 = 59

MULTI-STEP EQUATIONS MACC.912.A-CED.A.1: Create equations and inequalities in one variable and use them to solve problems.


method.


4 I am able to

• use multi-step equations to solve real-world applications or more challenging problems that I have never previously attempted

3 I am able to • solve multi-step equations in one variable

2 I am able to

• solve multi-step equations in one variable with help


TARGET


c) 3f + 2(f + 4) = 33 d) 4 − 2(3x + 9) = 28 EXAMPLE 2: SOLVING MULTI-STEP EQUATIONS WITH FRACTIONS AND DECIMALS

Solve.

a)

3h5

− h3= −2 b)

2m7

− 3m14

= 1

c) 2.5− 5d = −11.25 d) 1.2 = 2.4 − 0.6x

EXAMPLE 3: SOLVING EQUATIONS FOR REAL WORLD SITUATIONS

There is a 12-foot fence on one side of a rectangular garden. The gardener has 44 feet of fencing to enclose the other three sides. What is the length of the garden’s longer dimension?


Circle one: 4 3 2 1


WARM UP Describe and correct the error.

2x3

− 5 = 13

3 ⋅ 2x3

− 5⎛⎝⎜

⎞⎠⎟= 1

3

⎛⎝⎜

⎞⎠⎟⋅3

2x − 5 = 1

2x = 6

x = 3


Steps for Solving Equations with Variables on Both Sides • Simplify each side of the equation if possible (distribute, combine like terms).

If fractions exist, multiply by the LCD or distribute. • Add or subtract to get variables on one side and numbers without variables

on the other side of the equation • Multiply or divide to isolate the variable • Check answer

__________________________ - an equation is true for any value (Always)

__________________________ – an equation where there is no value to satisfy an equation (Never)

EXAMPLES

EXAMPLE 1: SOLVING EQUATIONS WITH VARIABLES ON EACH SIDE

Solve. a) 8 + 5y = 7y − 2 b) −3c − 12 = −5+ c

SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES MACC.912.A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by

letters. MACC.912.A-REI.A.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the

previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.


4 I am able to

• solve equations with variables on both sides that apply to real-world situations or more challenging problems that I have never previously attempted

3 I am able to • solve equations with variables on both sides

2 I am able to

• solve equations with variables on both sides with help


TARGET


c)

43

x + 2 = −7 + 13

x d)

12

x − x = 6 + 32

x

EXAMPLE 2: SOLVING EQUATIONS WITH GROUPING SYMBOLS

Solve. a) 5(y − 4) = 7(2y + 1) b) 12− 2(g − 7) = 3g + 3(g + 4)

c)

13

18 + 12t( ) = 6 2t − 7( ) d)

12

b − 2( ) = 16

2b + 4( )

EXAMPLE 3: FINDING SPECIAL SOLUTIONS

Solve. a) 2x + 8 = 5(x − 7)− 3x b) 6(q − 3)− 10 = 6q − 28

EXAMPLE 4: WRITING EQUATIONS

Five times the sum of a number and 3 is the same as 3 multiplied by 1 less than twice the number. What is the number? Write an equation and solve.


Circle one: 4 3 2 1


WARM UP Evaluate each expression for the given values of the variables.

1) x + 3y; x = 1, y = 3 2) x2 – y + xy; x = 2, y = –4


___________________________________ - an equation that uses at least 2 letters as variables. You can solve for any variable “in terms of” the other variables.

EXAMPLES

EXAMPLE 1: REWRITING A LITERAL EQUATION

Solve the following for the variable specified. a) x + y = 3, for y b) 3x – 4h = g, for x

c) 4x + 2y = 6, for y d) Hg = j + a

4

⎛⎝⎜

⎞⎠⎟

for a.

LITERAL EQUATIONS MACC.912.A-CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.


4 I am able to

• use formulas to solve real-world problems or to solve more challenging problems that I have never previously attempted

3 I am able to • solve equations for given variables

2 I am able to

• solve equations for given variables with help

1 I am able to • understand that equations may contain more than one variable.

TARGET


e) v = ⅓whl, for w f)

2+ xg

= 3y for g

EXAMPLE 2: LITERAL EQUATIONS FOR REAL WORLD SITUATIONS

A car’s fuel economy E (miles per gallon) is given by the formula E =m/g, where m is the number of miles driven and g is the number of gallons of fuel used.

a) Solve the formula for m.

b) If Claudia’s car has an average fuel consumption of 30 miles per gallon and she used 9.5 gallons, how far did she drive?

EXAMPLE 3: REWRITING A GEOMETRIC FORMULA

The formula for the volume of a cylinder is V = πr2h, where r is the radius of the cylinder and h is the height. a) Solve the formula for h.

b) What is the height of a cylindrical swimming pool that has a radius of 12 feet and a volume of 1810 cubic feet? Use 3.14 for pi.


Circle one: 4 3 2 1


WARM UP Simplify each product.

1)

407

× 218

2)

113× 24

66 3)

92× 36

27


A ___________________ is a comparison of two numbers by division.

The ratio x to y can be expressed as:

x to y x:y x/y

A _____________________ is a ratio with different units of measure, such as price per pound, miles per hour, etc.

A rate with a denominator of 1 is called a _______________________.

A ____________________________ is when two ratios are equal.

In order to check this, reduce both fractions to lowest form and check if they are the same.

Another way to check this is to use cross products. If the cross products are equal, then a proportion exists.

If

ab= c

d , then ad = bc

EXAMPLES

EXAMPLE 1: DETERMINING WHETHER RATIOS FORM PROPORTIONS

Determine whether the ratios form proportions.

a)

78

,4956

b)

0.250.6

,1.25

2 c)

45

,1620

RATIOS AND PROPORTIONS MACC.912.A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by

letters. MACC.912.N-Q.A.2: Define appropriate quantities for the purpose of descriptive modeling.


4 I am able to

• compare ratios and solve proportions in real-world situations or to solve more challenging problems that I have never previously attempted

3 I am able to

• compare ratios • solve proportions

2 I am able to

• compare ratios with help • solve proportions with help

1 I am able to • understand the definition of a ratio, rate, and proportion

TARGET


EXAMPLE 2: SOLVING A PROPORTION

Solve.

a)

x12

= 38

b)

212x

= 710

c)

h + 510

= 4 − h3

d)

x − 12x + 3

= 1211

EXAMPLE 3: SOLVING A REAL-WORLD SITUATION INVOLVING A PROPORTION

Jeff rides a 20-mile trail every Saturday. It takes him 4 hours. At this rate, how far can he ride in 7 hours? Set up a proportion and solve. (Let m represent miles)

EXAMPLE 4: SOLVING WITH SCALE MODELS

In a road atlas, the scale for the map of Connecticut is 5 inches = 41 miles. The scale for the map of Texas is 5 inches = 144 miles.

What are the distances in miles represented by 2

23

inches on each map? Set up proportions and solve.


Circle one: 4 3 2 1


WARM UP

Solve each equation.

1) 3x – (x – 4) = 2x 2) 4 + 6c = 6 – 4c


THE PERCENT PROPORTION

“A is P percent of B” can be represented by

AB= P

100 and B ≠ 0

where A is a part of the whole and B is the whole and also called the base.

P is the percent.

THE PERCENT EQUATION

“A is P percent of B” can be represented by

A = P% ⋅B and B ≠ 0

where A is a part of the whole and B is the whole and also called the base.

P is the percent as decimal.

SOLVING PERCENT PROBLEMS

Problem Type Example Proportion Equation

Find a percent What percent of 4 is 1.5?

1.54

= p100

1.5 = p% ⋅4

Find a part What is 25% of 200?

a200

= 25100

a = 25% ⋅200

Find a base 40% of what number is 12?

12b

= 40100

12 = 40% ⋅b

PERCENTAGES MACC.912.N-Q.A.3: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and

interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.


4 I am able to

• solve percent problems in real-world situations or more challenging problems that I have never previously attempted

3 I am able to

• solve percent problems

2 I am able to • solve percent problems with help

1 I am able to

• understand that one can solve a percent problem in a variety of ways

TARGET


EXAMPLES

EXAMPLE 1: FINDING A PERCENT

Find each percent. a) What percent of 350 is 28? b) What percent of 63 is 18?

c) What percent of 80 is 52? d) What percent of 200 is 210?

EXAMPLE 2: FINDING A PART

Find each percent. a) What is 12% of 80? b) What is 78% of 245?

c) What is 98% of 200? d) What is 3% of 15?

EXAMPLE 3: FINDING A BASE

Find each base. a) 45% of what number is 43.2? b) 56% of what number is 44.8?

c) 125% of what number is 32? d) 12% of what number is 240?

EXAMPLE 4: SOLVING A REAL-WORLD SITUATION INVOLVING A PERCENTAGE

$88 tickets to the Miami Dolphins game are offered at a 15% discount. What is the amount of the discount?


Circle one: 4 3 2 1


WARM UP

Solve each percent problem.

1) What percent of 13 is 100? 2) What is 30% of 5? 3) 15 is what percent of 45?


_________________________ - expresses an amount of change as a percent of an original amount.

If the new amount is greater than the original, then the percent change is called a

__________________________________. If the new amount is less than the original, then the percent change is

called a __________________________________.

PERCENT CHANGE

P% = amount of increase or decrease

original amount

• amount of increase = new amount – original

• amount of decrease = original amount – new amount

EXAMPLES

EXAMPLE 1: DETERMINING WHETHER EACH PERCENT CHANGE IS INCREASE OR DECREASE

Determine whether each percent change is an increase or decrease. Then find the percent change. a) original amount: 6 b) original amount: 105

new amount: 10 new amount: 95

c) original amount: 35 d) original amount: 45.5 new amount: 27 new amount: 65

PERCENT OF CHANGE MACC.912.N-Q.A.3: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and

interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.


4 I am able to

• find the percent of change in real-world situations or more challenging problems that I have never previously attempted

3 I am able to

• find the percent of change

2 I am able to • find the percent of change with help

1 I am able to

• understand the definition of percent of change

TARGET


EXAMPLE 2: FINDING A PERCENT DECREASE

a) A dress is on sale. The original price is $130. The sale price is $97.50. What is the discount expressed as a percent change?

b) A video game is on sale for $50. The original price was $65. What is the discount expressed as a percent change?

EXAMPLE 3: FINDING A PERCENT INCREASE

a) A music store buys a saxophone for $2800. The store then marks up the price of the saxophone to $3500. What is the mark up expressed as a percent change?

b) There were 1752 students that attended University High School last year. This year, 1925 students are attending. What is the percent of increase?

EXAMPLE 4: CALCULATING SALES TAX

Emma is purchasing a new car for $23,595 before tax. If the tax is 7% of the total sale, what is the final cost of the car?

EXAMPLE 5: DETERMINING FINAL PRICE

Lucius clipped a 35% off coupon in the Sunday paper for a new suit. The original price for the suit is $450 and sale tax is 6.5% of the discounted price. What is the final cost of the suit?


Circle one: 4 3 2 1

alg1 guided notes - unit 2 - solving equations

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