# Using Linear Equations to Solve Problems

1. Sam needs to get carpet for two rooms of his house. Estimate the number of square feet (square footage) in the two rooms if one room measures 10 ½ feet by 9 ¾ feet; and the other room measures 19¼ feet by 18 ½ feet.

2. Sandra drove for 234.8 miles and used 12.6 gallons of gas. Estimate the number of miles Sandra's car gets per gallon.

3. Melissa wants to have concrete driveway in front of her house. The driveway will measure 62 1/3 feet long and 19 feet wide; and will be 2/3 foot thick. How many cubic yards of cement will be needed for the driveway? Round your answer to the nearest cubic yard. [1 cubic yd = 27 cubic feet]

4. Use inductive reasoning to determine the next three numbers in the pattern: 3, 12, 26, 45, .....

5. Find a counterexample to show that the following statement is incorrect:

"The sum of any two odd numbers is divisible by 4"

6. An operator assisted call to another country costs $5.84 for 10 minutes. The same call dialed directly costs $0.39 for the first minute and $0.25 for each additional minute. How much money is saved on a 10 minute call by dialing directly?

7. Explain what a conjecture is. Which type of reasoning is used to arrive at a conjecture? Which type of reasoning is used to prove a conjecture? Provide an example.

8. Complete the magic square below (Please see attached document).

9. Sam found a number of nickels, dimes, and quarters in his room. He found three more dimes than nickels but twice as many quarters as dimes. The total value of the coins was $5.05. how many coins of each type did Sam find?

#### Solution Preview

1. Solution: The area of the first room is 10 ½ x 9 ¾=10.5 x 9.75=102.375 square feet

Area of the second room is 19¼ x 18 ½ =19.25 x 18.5=356.125 square feet.

Total area is 102.375+356.125=458.5 square feet.

2. Solution: 234.8 miles were covered using 12.6 gallons of gas. Hence the number of miles covered per gallon is given by (234.8)/12.6=18.63 miles(approximated to two decimals).

3. Solution: The quantity of cement needed is equal to the volume of the driveway. The volume is given by 62 1/3 x 19 x 2/3 cubic feet. Volume=831.77 cubic feet. In order to convert these cubic feet into cubic yards, we need to divide it further by 27 which give 30.806 cubic yards.

4. Solution: Observe ...

#### Solution Summary

Several word problems are solved using the method of solving linear equations. A detailed explanation for conjecture is given in one of the questions and question 8 is solved in an attached Word document. All steps are provided for reaching the final values for each problem.