Section 6.4 Applications of Linear Equations in One Variable

What You Will Learn Solving application problems involving linear equations

Translating Words to Expressions Phrase Mathematical Expression Six more than a number x + 6 A number increased by 3 x + 3 Four less than a number x – 4 A number decreased by 9 x – 9 Twice a number 2x Four times a number 4x

Translating Words to Expressions Phrase Mathematical Expression 3 decreased by a number 3 – x The difference between a number and 5 x – 5 Four less than 3 times a number 3x – 4 Ten more than twice a number 2x + 10

Translating Words to Expressions Phrase Mathematical Expression The sum of 5 times a number and 3 5x + 3 Eight times a number, decreased by 7 8x – 7 Six more than a number is 10. x + 6 = 10

Translating Words to Expressions Phrase Mathematical Expression Five less than a number is 20. x – 5 = 20 Twice a number, decreased by 6 is 12. 2x – 6 = 12 A number decreased by 13 is 6 times the number. x – 13 = 6x

To Solve a Word Problem 1. Read the problem carefully at least twice to be sure that you understand it. 2. If possible, draw a sketch to help visualize the problem. 3. Determine which quantity you are being asked to find. Choose a letter to represent this unknown quantity. Write down exactly what this letter represents.

To Solve a Word Problem 4. Write the word problem as an equation. 5. Solve the equation for the unknown quantity. 6. Answer the question or questions asked. 7. Check the solution.

Example 1: House Cleaning Service A professional house cleaning service charges a base fee of $20 plus $32.50 per hour to clean the Munoz’s house. If the Munoz’s total bill for house cleaning services from this company was $117.50 before taxes, how many hours did it take to clean their house?

Example 1: House Cleaning Service Solution Let n = number of hours $32.50n = cost for cleaning n hours at $32.50 per hour Base charge + hourly charge = total bill $20 + $32.50n = $117.50

Example 1: House Cleaning Service Solution It took 3 hours to clean the Munoz’s house.

Example 2: Dividing Land Mark IV Construction Company purchased 100 acres of land to be split into three parcels of land on which to build houses. One parcel of land will be three times as large as each of the other two. How many acres of land will each parcel contain?

Example 2: Dividing Land Solution Let x = # of acres in the 1st parcel x = # of acres in the 2nd parcel 3x = # of acres in the 3rd parcel So x + x + 3x = total amount of land x + x + 3x = 100 5x = 100 x = 20 Two parcels will contain 20 acres of land, the third will contain 60 acres.

Example 3: Dimensions of an Exercise Area Dr. Christine Seidel, a veterinarian, wants to fence in a large rectangular region in the yard behind her office for exercising dogs that are boarded overnight. She has 130 ft of fencing to use for the perimeter of the region. What should the dimensions of the region be if she wants the length to be 15 ft greater than the width?

Example 3: Dimensions of an Exercise Area Solution P = 2l + 2w, p is perimeter, l is length, w is width Let w = width l = w + 15 P = 300

Example 3: Dimensions of an Exercise Area Solution The width of the region is 25 ft and the length of the region is 40 ft.

Example 4: Craft Show Kim Stappenbeck is selling her homemade jewelry at a craft show. Determine the cost of a necklace before tax if the total cost of a necklace, including an 8% sales tax, is to be $34.56.

Example 4: Craft Show Solution Let x = cost of necklace before tax 0.08x = 8% of the cost of the necklace Cost of necklace before tax + tax on the necklace = 34.56 The cost of the necklace before tax is $32.