SOLVING SOLVING EQUATIONS AND EQUATIONS AND

PROBLEMSPROBLEMS

SOLVING SOLVING EQUATIONS AND EQUATIONS AND

PROBLEMSPROBLEMSCHAPTER 3CHAPTER 3

Section 3-1 Section 3-1 Transforming Transforming Equations: Equations:

Addition and Addition and SubtractionSubtraction

Section 3-1 Section 3-1 Transforming Transforming Equations: Equations:

Addition and Addition and SubtractionSubtraction

Addition Property of Equality

If a, b, and c are any real numbers, and a = b, then

a + c = b + c andc + a = c + b

Subtraction Property of

EqualityIf a, b, and c are any real numbers, and a = b, then

a - c = b - c andc - a = c - b

Equivalent Equations

Equations having the same solution set over a given domain.

-5 = n + 13 and -18 = n are equivalent

Transforming an Equation into an

Equivalent Equation

Transformation by Substitution

Substitute an equivalent

expression for any expression in a given equation.

Transformation by Addition

Add the same real number to each side of a given equation.

Transformation by Subtraction

Subtract the same real number from

each side of a given equation.

EXAMPLES

Solve:x – 8 = 17Add 8

x – 8 + 8 = 17 + 8x = 25

EXAMPLES

Solve:-5 = n + 13Subtract 13 -5 -13 = n + 13 – 13

-18 = n

EXAMPLES

Solve:x + 5 = 9Subtract 5

x + 5 – 5 = 9 - 5 x = 4

Section 3-2Section 3-2 Transforming Equations: Transforming Equations:

Multiplication and Multiplication and DivisionDivision

Section 3-2Section 3-2 Transforming Equations: Transforming Equations:

Multiplication and Multiplication and DivisionDivision

Multiplication Property of Equality

If a, b, and c are any real numbers, and a = b, then

ca = cb andac = bc

Division Property of Equality

If a and b are real numbers, c is any nonzero real number, and a = b, then

a/c = b/c

Transformation by Multiplication

Multiply each side of a given equation by the same nonzero

real number.

Transformation by Division

Divide each side of a given equation by the same nonzero

real number.

EXAMPLES

Solve:• 6x = 222• 8 = -2/3t• m/3 = -5

Section 3-3 Section 3-3 Using Several Using Several

TransformationTransformationss

Section 3-3 Section 3-3 Using Several Using Several

TransformationTransformationss

Inverse Operations

For all real numbers a and b,

(a + b) – b = a and(a – b) + b = a

Inverse Operations

For all real numbers a and all nonzero real numbers b

(ab) b = a and(a b)b = a

EXAMPLESSolve:1. 5n – 9 = 712. 1/5x + 2 = -13. 40 = 2x + 3x4. 8(w + 1) – 3 = 48

3-4 Using 3-4 Using Equations to Equations to

Solve ProblemsSolve Problems

3-4 Using 3-4 Using Equations to Equations to

Solve ProblemsSolve Problems

EXAMPLESThe sum of 38 and twice a number is 124. Find the number.

EXAMPLESThe perimeter of a trapezoid is 90 cm. The parallel bases are 24 cm and 38 cm long. The lengths of the other two sides are consecutive odd integers. What are the

lengths of these other two sides?

Solution

38

24

x + 2 x

3-5 Equations with 3-5 Equations with Variables on Both Variables on Both

SidesSides

3-5 Equations with 3-5 Equations with Variables on Both Variables on Both

SidesSides

EXAMPLES

•6x = 4x + 18•3y = 15 – 2y•(4 + y)/5 = y•3/5x = 4 – 8/5x•4(r – 9) + 2 = 12r + 14

3-6 Problem 3-6 Problem Solving: Using Solving: Using

ChartsCharts

3-6 Problem 3-6 Problem Solving: Using Solving: Using

ChartsCharts

PROBLEMA swimming pool that is 25 m long is 13 m narrower than a pool that is 50 m long. Organize in chart form.

SOLUTION

Length Width

1st pool

25 w -13

2nd pool

50 w

PROBLEMA roll of carpet 9 ft wide is 20 ft longer than a roll of carpet 12 ft wide. Organize in chart form.

SOLUTION

Width Length

1st roll 9 x + 20

2nd roll

12 x

PROBLEMAn egg scrambled with butter has one more gram of protein than an egg fried in butter. Ten scrambled eggs have as much protein as a dozen fried eggs.

How much protein is in

one fried egg?

SOLUTION

Protein per egg

Number of eggs

Total Protein

Scrambled egg

x + 1 10 10(x + 1)

Fried egg

x 12 12(x)

3-7 Cost, Income, 3-7 Cost, Income, and Value and Value ProblemsProblems

3-7 Cost, Income, 3-7 Cost, Income, and Value and Value ProblemsProblems

Formulas•Cost = # of items x price/item•Income = hrs worked x wage/hour•Total value = # of items x value/item

PROBLEMTickets for the senior class play cost $6 for adults and $3 for students. A total of 846 tickets worth $3846 were sold. How many student tickets were sold?

SOLUTION

number Price per ticket

Total Cost

Student

x 3 3x

Adult 846 - x 6 6(846-x)

PROBLEM

Marlee makes $5 an hour working after school and $6 an hour working on Saturdays. Last week she made $64.50 by working a total of 12 hours. How many hours did she work on Saturday?

SOLUTION

hours

wages

Income

Saturdays

x $6 6x

Weekdays

12-x $5 5(12-x)

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