Writing & Solving Equations

SOMETHING YOU MIGHT SAY…

In order to solve application problems, it is necessary to

translate English phrases into algebraic symbols. The following are some common phrases and their mathematic translation.

Applications

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

Applications

Translating from Words to Mathematical Expressions

Verbal Expression

4 less than a number

Mathematical Expression

(where x and y are numbers)

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

Applications

Translating from Words to Mathematical Expressions

Verbal Expression

14 times a number

Mathematical Expression

(where x and y are numbers)

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

Applications

Translating from Words to Mathematical Expressions

2

x

Verbal Expression

The quotient of 6 and a number

Mathematical Expression

(where x and y are numbers)

Division

A number divided by 15

half a number

(x ≠ 0)6x

x15

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 .

Applications

Caution

x15x

y

Applications

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 =.

Applications

Translating Words into Equations

Verbal Sentence Equation

16x – 25 = 87If the product of a number and 16 is decreased

by 25, the result is 87.

= 48The quotient of a number and the number plus

6 is 48.x + 6

x

+ x = 54The quotient of a number and 8, plus the

number, is 54.8x

Twice a number, decreased by 4, is 32. 2x – 4 = 32

Applications

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.

Six Steps to Solving Application Problems

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

Step 2 If possible draw a picture or diagram to help visualize the problem.

Step 3 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 4 Write an equation using the variable expression(s).

Step 5 Solve the equation.

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

Applications

Six Steps to Solving Application Problems

Now Lets Write & Solve Some Equations

Example 1:

Fifteen more than twice a number is – 23.

Now Lets Write & Solve Some Equations

Example 2:

The quotient of a number and 9, increased

by 10 is 11.

Now Lets Write & Solve Some Equations

Example 3:

The difference between 5 times a number

is 4 and 16.

CARTOON

Applications

Solving a Geometry Problem

Step 1 Read the problem. We must find the length and width of the rectangle.

The length is 2 ft more than three times the width and the perimeter is

124 ft.

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.

Step 2 Assign a variable. Let W = the width; then 2 + 3W = length.

Make a sketch.

W

2 + 3W

Step 3 Write an equation. The perimeter of a rectangle is given by the

formula P = 2L + 2W.

124 = 2(2 + 3W) + 2W Let L = 2 + 3W and P = 124.

Applications

Solving a Geometry Problem

Step 4 Solve the equation obtained in Step 3.

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.

124 = 2(2 + 3W) + 2W

124 = 4 + 6W + 2W

124 – 4 = 4 + 8W – 4

120 8W8 8

15 = W

Remove parentheses

124 = 4 + 8W Combine like terms.

Subtract 4.

Divide by 8.

120 = 8W

=

Applications

Solving a Geometry Problem

Step 5 State the answer. The width of the rectangle is 15 ft and the length is

2 + 3(15) = 47 ft.

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.

Step 6 Check the answer by substituting these dimensions into the words of

the original problem.