SOLVING WORD PROBLEMS USING LINEAR MODELS

Plan for Solving a Word Problem :

(i) Find out what numbers are asked for from the given information.

(ii) Choose a variable to represent the number(s) described in the problem. Sketch or a chart may be helpful.

(iii) Write an equation that represents relationships among the numbers in the problem.

(iv)  Solve the equation and find the required numbers.

(v) Answer the original question. Check that your answer is reasonable.

A linear function y = mx + b can be used as a model for many types of real life word problems which involve a constant rate of change.

Example 1 : 

A person travels home from work at a constant speed. Ten minutes after leaving work he is 20 miles from home, and 20 minutes after leaving work he is 12 miles from home. If he continues to travel at the same speed, how long will it take him to arrive home from work ?

Solution :

The problem asks for the number of minutes it takes to travel from work to home.

Start with the linear equation y = mx + b, in which y is the distance in miles from home, and x is the time in minutes.

When x = 10, y = 20

When x = 20, y = 12

First Equation : 

20 = 10m + b

Second Equation : 

20 = 10m + b

By subtracting the second equation from the first equation we get

8 = -10m

Divide each side by -10.

8/(-10) = m

-4/5 = m

Substitute m = -4/5 into the first equation. 

20 = 10(-4/5) + b

20 = -8 + b

Add 8 to each side.

28 = b

In y = mx + b, replace m with -4/5 and b with 28.

y = -4x/5 + 28

When he arrives home, y = 0. 

0 = -4x/5 + 28

Solve for x. 

4x/5 = 28

Multiply each side by 5/4.

x = 35

It takes 35 minutes from work to home.

Example 2 : 

At the beginning of a trip, the tank of Chloe’s car was filled with 12 gallons of gas. When she travels constantly on the highway 60 miles per hour, the car consumes 1 gallon of gas per 35 miles. If she traveled 5 hours and 15 minutes on the highway with a constant speed of 60 miles per hour, how many gallons of gas are left in the tank?

Solution :

Start with the linear equation y = mx + b, in which y is the number of gallons gas left in the tank after x miles of distance travelled.

m = change in y/change in x

It is given that the car consumes 1 gallon of gas for every 35 miles of distance. That is, amount of gas left in the tank is decreasing at the rate of 1 gallon per 35 miles.

m = -1/35

y = mx + b

y = (-1/35)x + b

y = -x/35 + b

Here, b is the amount of gas in the tank initially, that is, 12 gallons.

 y = -x/35 + 12 ----(1)

Distance travelled in 5 hours 15 minutes :

= speed x time

= 60 x 5¼

= 60 x 21/4

= 15 x 21

= 315 miles

Substitute x = 315 in (1).

 y = -315/35 + 12

y = -9 + 12

y = 3

After Chloe traveled 5 hours and 15 minutes, amount of gas left in the tank is 3 gallons.

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