Solving Verbal Solving Verbal EquationsEquations

Warm-upWarm-up

Translate the following into mathematical Translate the following into mathematical expressionsexpressions One number multiplied by the sum of two different One number multiplied by the sum of two different

numbersnumbers The product of three different numbers decreased by The product of three different numbers decreased by

a fourth numbera fourth number A number subtracted from the product of two different A number subtracted from the product of two different

numbersnumbers The difference between two numbers multplied by a The difference between two numbers multplied by a

third number. third number.

Warm-up ContinuedWarm-up Continued

Translate the following into mathematical Translate the following into mathematical expressionsexpressions The sum of a number and seven divided by a The sum of a number and seven divided by a

different numberdifferent number The product of fourteen and a number, The product of fourteen and a number,

decreased by another numberdecreased by another number Seven less than a number is nineteen.Seven less than a number is nineteen. A number increased by twelve is the same as A number increased by twelve is the same as

ten added to twice that number. ten added to twice that number.

Verbal EquationsVerbal Equations

Product ofProduct of Is the same asIs the same as QuotientQuotient Decreased byDecreased by More thanMore than isis

Increased byIncreased by Divided byDivided by Added toAdded to LessLess The quantity ofThe quantity of The differenceThe difference

What operation does the phrase tell me to do?

How to solve Verbal EquationsHow to solve Verbal Equations

Step 1: Write the algebraic expressionStep 1: Write the algebraic expression

Step 2: Solve for the variable Step 2: Solve for the variable

Step 3: Check your answerStep 3: Check your answer

Example 1Example 1

Fifteen is added to two times a number Fifteen is added to two times a number and the result is fifty-five.and the result is fifty-five.

2 15 55n Write the algebraic equation

Solve for the variable

2 40

20

n

n

2 20 15 55

40 15 55

55 55

Check your answer.

Example 2Example 2

Two consecutive integers sum up to 303. Two consecutive integers sum up to 303. Find the integers. Find the integers.

What do I use to represent consecutive integers again? 1 303n n

2 1 303n 2 302

151

n

n

Am I done?

Example 3Example 3

If three times a number is decreased by If three times a number is decreased by thirty, the result is equal to two times the thirty, the result is equal to two times the number. Find the number. number. Find the number.

3 30 2n n 30 0n 30n Am I done?

Example 4Example 4

The last math test that you took had 100 regular The last math test that you took had 100 regular points and 10 bonus points. You received a total points and 10 bonus points. You received a total score of 87, which included 9 bonus points. What score of 87, which included 9 bonus points. What would your score have been without any bonus would your score have been without any bonus points?points?

First Step: Write a model in words indicating what you have.

Regular points

Bonus points

Total score

Step 2: Fill in what you know

9 87p

Step 3: Solve for your variable

78p Are you done?

Step 4: Check your answer

Example 5Example 5

The perimeter of a rectangular shaped lot is The perimeter of a rectangular shaped lot is 420 meters. The length is twice the width. 420 meters. The length is twice the width. Find both dimensions.Find both dimensions. Step 1: draw a pictureStep 1: draw a picture

2x2x

xx

2x + x + 2x + x = 4202x + x + 2x + x = 420

x = 70x = 70 Am I done? Am I done?

width = 70, length = 140width = 70, length = 140

Example 6Example 6

Four people are sharing the cost of a monthly Four people are sharing the cost of a monthly phone bill of $58.25, what is each person’s phone bill of $58.25, what is each person’s share of the bill?share of the bill?

4 $58.25

$14.56

x

x

Does this cover the Does this cover the

whole bill?whole bill?

Example 7Example 7

The balance in your bank account The balance in your bank account is $642.35. If you use your check is $642.35. If you use your check card to buy some Halloween card to buy some Halloween decorations, your balance would be decorations, your balance would be $476.79. How much do the $476.79. How much do the decorations cost?decorations cost?

Consecutive IntegersConsecutive Integers

How do you represent consecutive How do you represent consecutive integers? Let’s try representing three integers? Let’s try representing three consecutive integersconsecutive integers

Why do I add by 1 each time?Why do I add by 1 each time?

x x+2x+1

Consecutive Even IntegersConsecutive Even Integers

How do you represent consecutive even How do you represent consecutive even integers? Let’s try three of them again. integers? Let’s try three of them again.

Why do I multiply the first by 2?Why do I multiply the first by 2? Why do I add by 2 each time?Why do I add by 2 each time?

2x 2x+42x+2

Consecutive Odd IntegersConsecutive Odd Integers

How do you represent consecutive odd How do you represent consecutive odd integers? Hey let’s try three yet again. integers? Hey let’s try three yet again.

Why did I multiply by 2 and add 1 the first Why did I multiply by 2 and add 1 the first timetime

Why do I add by 2 each time?Why do I add by 2 each time?

2x+1 2x+52x+3

Example 8Example 8

Find two consecutive integers that sum up Find two consecutive integers that sum up to fifty-three. to fifty-three.

1 53x x 2 1 53x 2 52

26

x

x

26 26 1 53

53 53

The two integers are 26 and 27

Example 9Example 9

Find three consecutive integers such that Find three consecutive integers such that the sum of the first and third integer is 82.the sum of the first and third integer is 82.

X + (X + 2) = 82X + (X + 2) = 82

2X + 2 = 822X + 2 = 82

2X = 802X = 80

X = 40X = 40

40, 41, 4240, 41, 42

Example 10Example 10

Find three consecutive even integers Find three consecutive even integers whose sum is 426. whose sum is 426.

2 2 2 2 4 426x x x 6 6 426x 6 420

70

x

x

The three integers are 140, 142, and 144

2(70) 2(70) 2 2(70) 4 426

140 140 2 140 4 426

426 429

Example 11Example 11 George is five cm taller than John. The George is five cm taller than John. The

sum of the heights is 405 cm. How tall is sum of the heights is 405 cm. How tall is George?George?

What are we looking for?Let George be our variable, x

Then define John in terms of George.

John = x-5

Then you can write your equation and solve

5 405x x 2 5 405

2 410

205

x

x

x

George is 205 cm.

205 205 5 405

410 5 405

405 405

Example 12Example 12

The Lions played twenty seven basketball The Lions played twenty seven basketball games. They won twice as many as they games. They won twice as many as they lost. How many did they win?lost. How many did they win?

Let the number they lost be x

Number they won =2x

2 27

3 27

9

x x

x

x

The Lions won 18 games.

Example 13Example 13

Find three consecutive odd integers Find three consecutive odd integers whose sum is 33whose sum is 33

9, 11, 139, 11, 13