CHAPTER 2

Solving Equations

*By:Hector Acosta* This chapter will help you learn how to solve equations, ratios, and proportions. Solving One-Step Equations Solving Two-Step Equations Solving Multi-Step Equations Literal Equations and Formulas Ratios, Rates, and Conversions Solving Equations With Variables on Both Sides Objective: Objective: Objective: Objectives: Objective: Objectives: Objectives: Solving Proportions Proportions and Similar Figures Percents Objectives: Objectives: Objective: To solve one-step equations in one variable To solve two-step equations Vocabulary Terms Equivalent equations

Addition Property of Equality

Subtraction Property of Equality

Isolate

Inverse operations

Multiplication Property of Equality

Division Property of Equality You can find the solution of an one-step equation by using the properties of equality and the inverse operations to write a simpler equivalent equations. Equivalent equations are equations that have the same solution. Inverse operations are operations that undo one another. They are four properties of equality. They are the Addition Property of Equality, Subtraction Property of Equality, Multiplication Property Of Equality, and Division Property of Equality. "Addition Property Of Equality" Addition Property of Equality is the property of equality that add the same number to each side of the equation. If the operation sign is a minus sign (-), then your using the addition property of equality. When using the Addition Property of Equality, you can add the same number to each side of an equation or isolate which makes an equivalent equation. Isolate is using properties of equality and inverse operations to get a variable witha a coefficient of 1 alone on one side of the equation. Example: x-5=8

(isolate) x-5+5=8+5

x=13 "Subtraction Property Of Equality" Subtraction Property of Equality is the property of equality in which you subtract the same number from each side of an equation. If the equation have a plus sign (+), you know you're gonna use the Subtraction Property of Equality When solving an equation with the Subtraction Property of Equality, You must subtract the same number from each side of an equation which makes an equivalent equation. Remember! In order to get variable by itself you must isolate it. Example: x+6=2

(isolate) x+6-6=2-6

x=-4 "Multiplication Property Of Equality" Multiplication Property of Equality is a property of equality in which you multiply each side of an equation. You can tell your using the Multiplication Property of Equality if the sign in the equation is this (/). When solving an equation with the Multiplication Property of Equality, you multiply each side of the equation by the same nonzero number which makes an equivalent equation. Example: x/6=12

(isolate) x/6*6=12*6

x=72 "Division Property Of Equality" Division Property of Equality is a property equality in which you divide each side of an equation. You would know that you're gonna use the Division Property of Equality if the multiplication sign (*) is in the equation. When using the Division Property of Equality, you would divide each side of the equation by the same nonzero number which makes an equivalent equation. Example: 3v=9

(isolate) 3v/3=9/3

v=3 Solving a two-step equation is a little different from solving a one-step equation. In a two-step equation, you are using the properties of equations more than once to isolate the variable. "Learning 2 Solve A Two-Step Equation" When solving a two-step equation, you must first identify the operations and the properties that your going to use. Also you must undo them using inverse operations. You can undo the operations in the reverse order of the order of operations. For example, to solve 5x-3=12, you can use addition first to undo the subtraction, and use division to undo the multiplication.

~Problem Workout~

5x-3=12

5x-3+3=12+3

5x=15

5x/5=15/5

x=3 "Using an Equation as a Model" This camp area already has 32 campsites around the lake. Now the park wants to add an equal number of sites to each of 30 acres. They want a total of 302 campsites. How many campsitess will they add to each acre?

~Number of acres:30 Equation: 30s+32=302

~Number of campsites per acre: ??? 30s+32-32=302-32

~Old campsites:32 30s=270

~Total number of campsites:302 30s/30=270/30

s=9 "Deductive Reasoning" When using deductive reasoning, you are stating your steps and the reasons for each step using the properties, definitions, or rules. For example, -k-7=4 STEPS REASONS -k-7=4

-k-7+7=4+7

-k=11

-1k=11

-1k/1=11

k=11 Original Equation

Addition Property Of Equality

Use addition to simplify

Multiplicative Property Of -1

Division Property Of Equality

Use division to simplify to solve multi-step equations with one variable "Solving Multi-Step Equations" To solve multi-step equations, you must use the properties of equality, inverse operations, and the properties of real numbers. When solving a multi-step equation, you must first combine all like terms.

For example: 10=2m-5+3m, first your going to combine the like terms, which is 2m and 3m. Then you continue the regular steps to isolate the variable. "Problem workout"

10=2m-5+3m

10=2m-5+3m

10=5m-5

10-5=5m-5

5=5m

5/5=5m/5

1=m "A Multi-Step Equation As A Model" John owns twice as many books as Kate owns. Adding 6 to the number of comic books James owns and then dividing by 7 give the number Cory owns. Cory owns 30 comic books. How many does Kim own? What do we know? Comic books James own: 2c

Comic books Kim own: "c"

Comic books Caden: (2*c)+6

7 =30 Equation were using

2c+6

7 =30 2c+6

7 =30 2c+6

7 =30*7 *7 2c+6=210

2c+6-6=210-6

2c=204 2 2 c =102 "Using the Distributive Property" What is the solution of 2(2p-3)=36? 2(p-3)=36

2p-6=36

2p-6+6=36+6

2p=42

2p/2=42/2

p=21 Original Equation

Distributive Property

Subtract 6 from each side

Simplify

Divide each side by 2

Simplify "An Equation That Contains A Fraction" What is the solution for 2x x

4 3 - =10 ? 2x 1x

4 3 - =10 6x 4x

12 12 - =10 2x

12 =10 12 2x

2 12 =(10) 12

2 x=60 Write a fraction using a common denominator, 12.

Combine like terms

Multiply each side by , the reciprocal of

Simplify 12

2 2

12 "An Equation That Contains A Decimal" What is the solution of 6.4-0.04p=2.24? 6.4-0.04p=2.24

100(6.4-00.4)=100(2.24)

640-4p=224

-640 =-640

-4p=-416

-4p/-4=-416/-4

p=104 Multiply each side by 100

Distributive Property

Subtract 640 from each side

Simplify

Divide each side by (-4)

Simplify To solve equations on both sides

To identify equations that are identities or have no solution "Solving Equations With Variables On Both Sides" To solve equations with variables on both sides, you can use the properties of equality and inverse operations to write a series of simpler equivalent equations. For example: 6x+3=2x+19

6x+3=2x+19

6x+3-2x=2x+19-2x

4x+3=19

4x+3-3=19-3

4x=16

4x/4=16/4

x=4 Original Equation

Subtract 2x from each side.

Simplify.

Subtract 3 from each side.

Simplify.

Divide each side by.

Simplify for answer. "Vocabulary Term" Identity "Solving An Equation With Grouping Symbols" What is the solution of 1(2x-3)=1(x+5)? 1(2x-3)=1(x+5)

2x-3=x+5

2x-3-x=x+5-x

x-3=5

x-3+3=5+3

x=8 Original Equation.

Distributive Property.

Subtract x from each .

Simplify.

Add 3 to each side.

Simplify to get answer. "Identities And Equations With No Solution" An equation that is true for every possible value of the variable is an identity. For example; x+5=x+5 is an identity. An equation has no solution if there is no value of the variable that makes the equation true. For example; x+6=x+9 has no solution. Identity 20x+30=10(2x+3)

20x+30=20x+30

20x+30-20x=20x+30-20x

30=30

30-30=30-30

0=0 Original Equation.

Distributive Property.

Subtract 20x from each side.

Simplify.

Subtract 30 from each side.

Simplify to get identity. Since 20x+30=20x+30 is always true, there are infinitely many solutions of the equation.The equation is a identity. Also, when you get to 0=0, the both sides are the same. No Solution 4m-2=2m+8+2m

4m-2=4m+8

4m-2-4m=4m+8-4m

-2=8 Original Equation

Combine Like Terms.

Subtract 4m from each side.

Simplify. Because -2=8(not equal to), the equation has no solution. "Using An Equation With Variables On Both Sides" Jake and Greg have the same number of action figures in their collections. Jake has 8 complete sets plus 3 individual figures and Greg has 4 complete sets plus 27 individual figures. How many figures are in a complete set? What Do You Know?

Jake has 8 sets plus 3 individual figures

Greg has 4 sets plus 27 individual figures

(n) represent number of figures in a complete set Equation :8n+3=4n+27 8n+3=4n+27

8n+3-4n=4n+27-4n

4n+3=27

4n+3-3=27-3

4n=24

4n/4=24/4

n=6 Original Equation

Subtract 4n from each side.

Simplify.

Subtract 3 from each side.

Simplify.

Divide each side by 4.

Simplify to get answer. To rewrite and use literal equations and formulas Vocabulary Terms Literal Equation

Formula "Working With Literal Equations" You will be learning to solve problems using equations with more than one variable. A literal equation is an equation that involves two or more variables. When working out a literal equation, you can use the methods you've learned to isolate any kind of variable. FOR EXAMPLE "Rewriting Literal Equation With Only Variables" When rewriting a literal equation with only variables, you should assume that the variable or variable expression is not equal to zero because division by zero is not defined. Also, you should treat the variables are not solving for as constants. What equation do you get when you solve ax+bx=c for x?

ax+bx=c

x(a+b)=c

x(a+b)= c a+b a+b x= c a+b Original Equation

Distributive Property

Divide each side by a-b

Simplify "Formula" A formula is an equation that states a relationship among quantities. Formulas are special types of literal equations.The ones below are some common formulas. Formula Name Formula Definitions of Variables Perimeter of rectangle P=2l+2w P=Perimeter, L=Length, W=Width

Circumference of a circle C=2 r C=Circumference, R=Radius

Area of rectangle A=lw A=Area, L=Length, W=Width

Area of triangle A=1/2bh A=Area, B=Base, H=Height

Area of a circle A= r2 A=Area, R=Radius

Distance Traveled d=rt D=Distance, R=Rate, T=Time

Temperature C=5/9(F-32) C=degrees Celsius, F=degrees Fahrenheit "Rewriting A Geometric Formula" What is the width of a rectangle that has a length of 6 and Perimeter of 30? Solve for w. 30=2(6)+2w

30=12+2w

30-12=12+2w-12

18=2w

18/2=2w/2

9=w Original Equation

Distribute

Subtract 12 from each side

Simplify

Divide each side by 2

Simplify to get answer "Rewriting A Formula" A group of dolphins took a journey to the Pacific Ocean. The distance that the group of dolphins travel was 3,000 miles. It takes a typical dolphin about 8 days to reach there. What is the average rate at which a dolphin travels in per day? Round the nearest mile per day. What is the appropriate formula? d=rt

d=rt

d/t=rt/t

d/t=r

3,000/8=r

375=r Formula

Divide each side by t

Simplify

Substitute 3,000 with d and 8 with t

Simplify Work Detail!!!!!! To find ratios and rates

To convert units and rates Vocabulary Terms Ratio

Rate

Unit Rate

Conversion Factor

Unit Analysis A ratio compares two numbers by division. Ratios could be written three ways: a/b, a:b, and a to b. An example of a ratio is, there are 7 boys to 2 girls on the swim team. A ratio that compares quantities measured in different units is called a rate. A rate with a denominator of 1 unit is a unit rate. You can write ratios and find unite rates to compare quantities. Also, you can convert unit rates to solve problems. "Comparing Unit Rates" To convert from one unit to another, you multiply the original unit by a conversion factor which makes the desired unit. A conversion factor is a ratio of two equivalent measures in different units. The conversion factor is always equal to 1. "Converting Units" In the video, Notice that the units for each quantity are included in the calculations to help determine the units for the answers. This process is called unit analysis. "Converting Rates" Rates compare measures in two different units, so you must multiply by two conversion factors to change both of the units. To solve and apply proportions Vocabulary Words Proportion

Cross products

Cross Products Property Oppa

Clarkston

Style Proportions A proportion is an equation that states that two ratios are equal. If two ratios are equal and a quantity in one of the ratios is unknown, you can write and solve a proportion to find the unknown quantitiy. "Solving A Proportion Using The Multipication Property" What is the solution of the proportion = ? 2 m

3 4 2 m

3 4

2 m

3 4 =

= 4* *4 8

3 =m 2.67=m Proportion

Multiply each side by 4.

Simplify.

Divide to get solution. "Cross Products" The products ab and cd are called cross products. You can use the cross products property to solve proportions. Cross Products Property is the property in which cross products of a proportion are equal. a c

b d

a c

b d

a c

b d =

=

= bd* *bd

bd* *bd da=cb

ad=bc Suppose this equation is true.

Multiplication Property of Equality

Divide the common factors Simplify.

Commutative Property of Multiplication "Solving A Proportion Using The Cross Products Property" What is the solution for the proportion = ? 5 10

6 y 5 10

6 y

5y=6(10)

5y=60

5y/5=60/5

y=12 = Proportion

Cross Product Property

Multiply to simplify.

Divide each side by 5.

Simplify to get solution "Solving A Multi-Step Proportion" What is the solution of the proportion = ? p-2 p+2

4 6 p-2 p+2

4 6

6(p-2)=4(p+2)

6p-12=4p+8

6p-12-4p=4p+8-4p

2p-12=8

2p-12+12=8+12

2p=20

2p/2=20/2

p=10 = Proportion

Cross Products Property

Distributive Property

Subtract 4p from each side.

Simplify.

Add 12 to each side.

Simplify Divide each side by 2.

Simplify for solution. To find missing lengths in similar figures

To use similar figures when measuring indirectly Vocabulary Terms Similar figures

Scale drawing

Scale

Scale model Similar figures have the same shape but not necessarily the same size. You can proportions to find missing lengths in similar figures. In similar figures, the measures of corresponding angles are equal, and corresponding side lengths are in proportion. The order of letters when you name similar figures is important. The reason for this is because it tells which parts of the figures are corresponding parts. For example, DEF~GHI, the following is true. "Finding The Length Of A Side" In the diagram, ABC~ DEF. What is DE? A C B D F E 10 10 5 5 BC AB

EF DE

10 10

5 DE

10(DE)=5(10)

10DE=50

10DE/10=50/10

DE=5 =

= Write a proportion

Substitute lengths

Cross Products Property

Multiply

Divide each side by 10

Simplify You can use similar figures and proportions to find lengths that you cannot measure directly. "Applying Similarity" 3ft 5ft A tree casts a shadow of 7.5 ft long. The woman is 5ft tall and cast a shadow of 3ft long. The triangle shown for the tree and its shadow is similar to the triangle shown for the woman and her shadow. How tall is the tree? 7.5ft x Workout 3 5

7.5 x

3(x)=7.5(5)

3x=37.5

3x/3=37.5/3

x=12.5 = Write a proportion

Cross Products Property

Multiply to simplify

Divide each side by 3

Simplify to get answer. A scale drawing is a drwaing that is similar to an actual object or place. Some examples of scale drawings are blueprints, floor plans, and maps. In a scale drawing, the ratio of any length on the drawing to the actual length is always the same. It is called the scale of a drawing. "Interpreting Scale Drawings" The scale of the map at the left is 1inch:10 miles. Approximately how far is it from Valkaria to Wabasso?

Map Distance=1.75 in. 1 1.75

10 d

1*d=10(1.75)

d=17.5 = Write proportion

Cross Products Property

Simplify Map

Actual A scale model is a three-dimensional model that is similar to a three-dimensional object. "Using Scale Models" To solve percent problems using proportions

To solve percent problems using the percent equation "The Percent Proportion" You can solve problems involving percents using either the percent proportion or the percent equation, which are closely related. If you write a percent as a fraction, you can use a proportion to solve a percent problem. When using the percent proportion, the proportion would look like this; . The a is p percent of b. a p

b 100 = "The Percent Equation" When you write as p% and solve for a, you get the equation a=p%*b.This equation is called the percent equation. You can use either the percent equation or the percent proportion to solve any percent problem. When using the percent equation, the equation would look like this a=p%*b and "What percent of b is a?". p

100 "Find A Part" A evening dress that normally cost $155 is on sale for 35% off. What is the sale price of the shirt? Step 1: Use the percent equation to find the amount of the discount.

a=p%*b Write the percent equation

=35%*155 Substitute 35 for p, 155 for b

=0.35*155 Write the percent as a decimal

=54.25 Multiply for amont of discount

Step 2: Find the sale price.

$155-$54.25= $100.75

The sale price of the evening dress is $100.75. "Finding The Base" 160% of what number is 18.5?

a=p%*b Write the percent equation

18.5=160%*b Substitute 18.5 for a, 160 for p

18.5=1.60*b Write the percent as a decimal

18.5/1.60=1.60*b/1.60 Divide each side by 1.60

11.56=b Simplify to gat answer. "Simple Interest Formula" A common application of percents is a simple interest, which is interest you earn on only the principal in an account. The simple interest formula is where I is interest, P is principal, r is the annual rate written as a decimal, and t is the time in years. It would look like this; I=Prt. Vocabulary Terms Percent change

Percent increase

Percent decrease

Relative error

Percent error "Percent Change" Percent change is the ratio of the amount of change to the original amount. A percent change expresses an amount of change as a percent of an original amount. You can find a percent change when you know the original amount and how much it has changed. Change Expressed as a Percent To find percent change

To find the relative error in linear and nonlinear measurements "Finding A Percent Increase and Percent Decrease" If a new amount is greater than the original amount, the percent change is called a percent increase. A example of finding a percent increase is a percent markup. If the new amount is less than the original amount, the percent change is called a percent decrease. An example of finding a percent decrease is finding a percent discount. "Relative Error" Relative error is a ratio of the absolute value of the difference of a measured value and an actual value compared to the actual value. When relative error is expressed as a percent, it is called percent error. THANK YOU FOR WATCHING!!!

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# ALGEBRA1~CHAPTER 2

My prezi on Chapter 2 Lessons 1-10. Hope U Enjoy!!!

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