Chapter 3

Solving Linear Equations

ByKristen Sanchez and Emma Nuss

INVERSE OPERATIONS:INVERSE OPERATIONS:• Two operations that undo each other, such Two operations that undo each other, such

as addition and subtraction are called as addition and subtraction are called inverse operations. These can help you inverse operations. These can help you isolate the variable on one side of the isolate the variable on one side of the equation.equation.

• To solve an equation with a fractional To solve an equation with a fractional coefficient, such as 10 = 2/3m, multiply coefficient, such as 10 = 2/3m, multiply each side of the equation by the reciprocal each side of the equation by the reciprocal of the fraction (in this case 3/2). The of the fraction (in this case 3/2). The product of a nonzero number and its product of a nonzero number and its reciprocal is 1.reciprocal is 1.

Summary: Properties of EqualitySummary: Properties of Equality

• Addition Property of Equality:Addition Property of Equality:If a = b, then a + c = b + cIf a = b, then a + c = b + c

• Subtraction Property of Equality: Subtraction Property of Equality: If a = b, then a – c = b – cIf a = b, then a – c = b – c

• Multiplication Property of Equality: Multiplication Property of Equality: If a = b, then ca = cbIf a = b, then ca = cb

• Division Property of Equality:Division Property of Equality:If a = b and c does not = 0, then a/c = b/cIf a = b and c does not = 0, then a/c = b/c

3.1: Solving Equations Using Addition and Subtraction

Operation Original Equation

Equivalent Equation

Add the same number to each side

x-3=5 x=8

Subtract the same number from each side

x+6=10 x=4

Simplify one or both sides

x=8-3 x=5

3.2: Solving Equations Using Multiplication and Division

Operation Original Equation

Equivalent Equation

Multiply each side of by the same nonzero number.

x/2=3 X=6

Divide each side of by the same nonzero number.

4x=12 X=3

Solving Multi-Step Equations

• When solving a multi-step equation, the first step is isolate the variable, then solve for x. Example:3x+7=-8 -7 -7 3x=-15 x=-5

Solving Equations with Variables on Both Sides

• Some equations have variables on both sides. To solve these, you collect the variable terms on one side of the equation.

• Example:7x+19=-2x+55+2x +2x

9x+19=55 -19 -19 9x=36 9 9 x=4

Tips on Solving Linear Equations

• Simplify: combine like terms or distribution• Collect: put the variable terms on the side

with the larger coefficient• Inverse operations: use inverse operations

to isolate the variable• Check: check your solution with the

original equation

3.6: Solving Decimal Equations3.6: Solving Decimal Equations• If you have a long decimal answer to an equation, it is If you have a long decimal answer to an equation, it is

often more practical to round to an approximate answeroften more practical to round to an approximate answer• Example:Example:

-38x-39=118-38x-39=118 +39 +39+39 +39-38x=157-38x=157 x=157/-38x=157/-38 xx≈-4.131578947≈-4.131578947 x≈-4.13x≈-4.13

• The rounded answer may not be exactly equal to the The rounded answer may not be exactly equal to the original equation, but the symbol ≈ is used to show that original equation, but the symbol ≈ is used to show that the answer approximately balances the two sides of the the answer approximately balances the two sides of the equation.equation.

3.7: Formulas 3.7: Formulas • Formulas are algebraic equations that relate two or more Formulas are algebraic equations that relate two or more

quantities. quantities. • Formulas can be used to solve problems with variables Formulas can be used to solve problems with variables

other than x and y.other than x and y.• Example:Example:

Celsius and Fahrenheit are related by the Celsius and Fahrenheit are related by the equation C=5/9(F-32) where C is Celsius and F is equation C=5/9(F-32) where C is Celsius and F is Fahrenheit. Solve the formula for degrees Fahrenheit F. Fahrenheit. Solve the formula for degrees Fahrenheit F.

C=5/9(F-32)C=5/9(F-32) *9/5 *9/5*9/5 *9/5 9/5C=F-329/5C=F-32 +32 +32+32 +32 9/5C+32=F9/5C+32=F

3.8: Ratios and Rates3.8: Ratios and Rates

• The ratio of a to b is a/b. The ratio of a to b is a/b. • If a and b are measured in different units, If a and b are measured in different units,

then a/b is called the rate of a per b.then a/b is called the rate of a per b.• Rates are often expressed as unit rates, or Rates are often expressed as unit rates, or

the rate per one given unit (ex. 60 miles the rate per one given unit (ex. 60 miles per 1 gallon)per 1 gallon)

Unit AnalysisUnit Analysis

• Writing the units when comparing each Writing the units when comparing each quantity of a rate is called unit analysis.quantity of a rate is called unit analysis.

• You can multiply and divide units just as You can multiply and divide units just as you can multiply and divide numbers. you can multiply and divide numbers.

• Example:Example:60 mins= 1 hour60 mins= 1 hourconvert 3 hours to minutesconvert 3 hours to minutes3 hours x 60 mins/1 hour= 180 mins3 hours x 60 mins/1 hour= 180 mins

3.9: Percents3.9: Percents• A percent is a ratio that compares a number to 100. For A percent is a ratio that compares a number to 100. For

example, forty percent can be written as 40/100, 0.40, or example, forty percent can be written as 40/100, 0.40, or 40%.40%.

• Number being compared to base= aNumber being compared to base= a percent= p/100percent= p/100 base number = bbase number = b a = p/100 * ba = p/100 * b• Example:Example:

Fourteen dollars is 25% of what amount of money?Fourteen dollars is 25% of what amount of money?Percent = 25/100 =1/4Percent = 25/100 =1/414 = ¼ * x14 = ¼ * x4(14) = 4(1/4) * x4(14) = 4(1/4) * x56 = x56 = x

Using Percents to find DiscountsUsing Percents to find Discounts• Discount is the difference between the regular price of Discount is the difference between the regular price of

and item and its sale price. To find the discount percent, and item and its sale price. To find the discount percent, use the regular price as the base number in the percent use the regular price as the base number in the percent equation.equation.

• ExampleExampleA portable CD player has a regular price of $90 A portable CD player has a regular price of $90

and a sale price of $72. What is the discount percent?and a sale price of $72. What is the discount percent?Discount=regular price-sale priceDiscount=regular price-sale price

=90-72=18=90-72=18Percent=p/100Percent=p/100Regular price=90Regular price=9018=p/100 * (90)18=p/100 * (90)18/90=p/10018/90=p/100.20=p/100.20=p/10020=p20=p

Summary of Percents

Question Given Need to Find

What is p percent of b?

b and p Number compared to base, a

a is p percent of what?

a and p Base number, b

a is what percent of b?

a and b Percent, p

a = p/100 * b