# WRITE VARIABLE EQUATIONS

When writing equations, we use variables to represent the unspecified numbers or measures referred to in the problem. Then write the verbal expressions as algebraic expressions.

Some verbal expressions that suggest the equals sign are listed below.

"is, is equal to, is as much as, equals, is the same as, is identical to"

We use the following four steps to represent to solve any word problem.

Step 1 :

Explore the problem :

To solve a verbal problem, first read the problem carefully and explore

• Identify what information is given.
• Identify what we are asked to find.

Step 2 :

Plan the solution :

One strategy we can use to solve a problem is to write an equation. Choose a variable to represent one of the  unspecific numbers in the problem.

This is called the defining a variable. Then use the variable to write expressions for the other unspecified numbers in the problem.

Step 3 :

Solve the problem :

Solve for unknown.

Step 4 :

Examine the solution :

Check answer in the context of the original problem.

• Does our answer make sense?
• Does it fit the information in the problem?

Example 1 :

The first ice cream plant was established in 1851 by Jacob Fussell. Today, 2,000,000 gallons of ice cream are produced in the United States each day. In how many days can 40,000,000 gallons of ice cream be produced in the United States?

Solution :

Step 1 :

Given information :

Quantity of ice cream to be produced.

What do we have to find ?

Number of days required to produce the given quantity of ice cream.

Step 2 :

Let T be the required number of days to produce 40,000,000 gallons of ice cream.

Quantity of ice cream produced in each day = 2,000,000 gallons.

2,000,000 x T  =  40,000,000 ----(1)

Step 3 :

Divide by 2,000,000 on both sides,

T  =  40,000,000/2,000,000

T  =  20

Step 4 :

Substitute T = 20 in (1).

2,000,000 x 20  =  40,000,000

40, 000, 000  =  40,000,000

Example 2 :

Misae has \$1900 in the bank. She wishes to increase her account to a total of \$3500 by depositing \$30 per week from her paycheck. Will she reach her savings goal in one year?

Solution :

Step 1 :

Given information :

Misae has \$1900 in the bank.

She wishes the total amount to be \$3500

She wants to deposit \$30 per week.

What do we have to find ?

Will she reach her savings goal in one year?

For this we need to find the number of weeks required to reach the total amount of \$3500.

Step 2 :

Let T be the required number of weeks

1900 + 30T  =  \$3500 ----(1)

Step 3 :

Subtract 1900 on both sides

1900 - 1900 + 30T  =  3500 - 1900

30T  =  1600

Divide by 30 on both sides

30T/30  =  1600/30

T = 53.33

Step 4 :

Substitute T = 53.33 in (1).

1900 + 30(53.33)  =  \$3500

1900 + 1599.9  =  \$3500

\$3499.99    \$3500

Example 3 :

Mrs. Patton is planning to place a fence around her vegetable garden. The fencing costs \$1.75 per yard. She buys f yards of fencing and pays \$3.50 in tax. If the total cost of the fencing is \$73.50, write an equation to represent the situation.

Solution :

Step 1 :

Given information :

Cost of fencing per yard  =  \$1.75

Tax = \$3.50

Total cost of fencing  =  \$73.50

What do we have to find ?

Total number of yards to be fenced around the vegetable garden.

Step 2 :

Let y be the required number of weeks

1.75y + 3.50  =  \$73.50

Step 3 :

Subtract 3.50 on both sides

1.75y + 3.50 - 3.50 =  73.50 - 3.50

1.75Y  =  70

Divide by 1.75 on both sides

1.75y/1.75  =  70/1.75

y  =  40

Step 4 :

Substitute y = 40 in (1).

1.75 (40) + 3.50  =  \$73.50

70 + 3.50  =  73.5

73.50  =  73.50

Example 4 :

Ken and Donna are driving from Baltimore, MD to Richmond, VA for a wedding. The total distance they must drive is 131 miles. Since the reception is late, they decided to get a bite to eat along the way. They want the first part of the trip to be 65 miles more than the second part. How many miles will they drive before they stop?

Solution :

Let x be the distance of the second part. Then distance covered is x + 65

Total distance to be covered = 131

x + 65 + x = 131

2x + 65 = 131

2x = 131 - 65

2x = 66

x = 66/2

x = 33

So, 33 miles has been covered before they stop.

Example 5 :

Ben sold his used snow plow and accessories for \$352. If he received seven times as much money for the snow plow as he did for the accessories, find how much money he received for the snow plow.

Solution :

Let x be the amount he has received for accessories.

Amount he has received for snow plow = 7x

7x + x = 352

8x = 352

x = 352/8

x = 44

Amount he has received for snow plow = 7(44)

= 308

So, he has received \$308 for snow plow.

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