Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 2.3 - 1 2.3 Applications of Linear Equations

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 2.3 - 1

2.3

Applications of Linear Equations

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 2

2.3 Applications of Linear Equations

Problem-Solving Hint

PROBLEM-SOLVING HINT

Usually there are key words and phrases in a verbal problem that translate

into mathematical expressions involving addition, subtraction, multiplication,

and division. Translations of some commonly used expressions follow.



Translating from Words to Mathematical Expressions

Verbal Expression

The sum of a number and 2

Mathematical Expression

(where x and y are numbers)

Addition

3 more than a number

7 plus a number

16 added to a number

A number increased by 9

The sum of two numbers

x + 2

x + 3

7 + x

x + 16

x + 9

x + y




Verbal Expression

4 less than a number



Subtraction

10 minus a number

A number decreased by 5

A number subtracted from 12

The difference between two

numbers

x – 4

10 – x

x – 5

12 – x

x – y




Verbal Expression

14 times a number



Multiplication

A number multiplied by 8

Triple (three times) a number

The product of two numbers

14x

8x

3x

xy

of a number (used with

fractions and percent)

34 x3

4




Verbal Expression

The quotient of 6 and a number



Division

A number divided by 15

The ratio of two numbers

or the quotient of two numbers

(x ≠ 0)6x

(y ≠ 0)xy

x15


CAUTION

Because subtraction and division are not commutative operations, be carefulto correctly translate expressions involving them. For example, “5 less than anumber” is translated as x – 5, not 5 – x. “A number subtracted from 12” isexpressed as 12 – x, not x – 12. For division, the number by which we are dividing is the denominator, andthe number into which we are dividing is the numerator. For example, “a number divided by 15” and “15 divided into x” both translate as . Similarly,“the quotient of x and y” is translated as .


Caution

x15x

y



Indicator Words for Equality

Equality

The symbol for equality, =, is often indicated by the word is. In fact, any

words that indicate the idea of “sameness” translate to =.



Translating Words into Equations

Verbal Sentence Equation

If the product of a number and 16 is decreased

by 25, the result is 87.

The quotient of a number and the number plus

6 is 48.

The quotient of a number and 8, plus the

number, is 54.

Twice a number, decreased by 4, is 32.



Distinguishing between Expressions

and Equations

(a) 4(6 – x) + 2x – 1

(b) 4(6 – x) + 2x – 1 = –15

There is no equals sign, so this is an expression.

Because of the equals sign, this is an equation.

Decide whether each is an expression or an equation.


Solving an Applied Problem

Step 1 Read the problem, several times if necessary, until you understandwhat is given and what is to be found.

Step 2 Assign a variable to represent the unknown value, using diagrams or tables as needed. Write down what the variable represents. Express any other unknown values in terms of the variable.

Step 3 Write an equation using the variable expression(s).

Step 4 Solve the equation.

Step 5 State the answer to the problem. Does it seem reasonable?

Step 6 Check the answer in the words of the original problem.


Six Steps to Solving Application Problems


Solving a Geometry Problem

The length of a rectangle is 2 ft more than three times the width. The perimeter

of the rectangle is 124 ft. Find the length and the width of the rectangle.


Finding Unknown Numerical Quantities

A local grocery store baked a combined total of 912 chocolate chip cookies

and sugar cookies. If they baked 336 more chocolate chip cookies than sugar

cookies, how many of each did the store bake?


Solving a Percent Problem

During a 2-day fundraiser, a local school sold 1440 raffle tickets. If they sold

350% more raffle tickets on the second day than the first day, how many raffle

tickets did they sell on the first day?


Solving an Investment Problem

A local company has $50,000 to invest. It will put part of the money in an

account paying 3% interest and the remainder into stocks paying 5%. If the

total annual income from these investments will be $2180, how much will be

invested in each account?


Solving a Mixture Problem

A chemist must mix 12 L of a 30% acid solution with some 80% solution to get

a 60% solution. How much of the 80% solution should be used?



Problem-Solving Hint

PROBLEM-SOLVING HINT

When pure water is added to a solution, remember that water is 0% of the

chemical (acid, alcohol, etc.). Similarly, pure chemical is 100% chemical.


Solving a Mixture Problem

A chemist must mix 8 L of a 10% alcohol solution with pure alcohol to get a

a 40% solution. How much of the pure alcohol solution should be used?

Documents

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 2.3 - 1 2.3 Applications of Linear Equations