Linear Equation Word Problems

Step 1: Read the problem.

Step 2: There should be one such quantity which you need to find. Represent that quantity in terms of a variable.

Step 3: Relate all other unknown quantities in terms of the variable assumed in step 2.

Step 4: Form an equation according to the information in question.

Step 5: Solve the equations to find the value of variables.

Example: The sum of three consecutive even numbers is 78. What is the smallest of these numbers?

Let us assume that the three consecutive even numbers are x, x + 2, and x + 4.

x + x + 2 + x + 4 = 78

3x + 6 = 78

3x = 72 implies x = 24

Check your answer: x + (x + 2) + (x + 4) = 24 + (24 + 2) + (24 + 4) = 24 + 26 + 28 = 78

Example: If the difference between two numbers is 38, and the ratio of two numbers is 6:5. Find the numbers.

Let the numbers be x and y.

Then x – y = 38 … Equation (1)

         x : y = 6 : 5 … Equation (2)

From equation (1), x = 38 + y

Equation (2) implies  =>  5x = 6y

Put x = 38 + y in 5x = 6y Þ 5(38 + y) = 6y

228 + 5y = 6y implies y = 228

x = 38 + 228 = 266

The numbers are 228 and 266.

Check Point

1. Denise bought a magazine for $2 and some notepads for $3 each. She spent a total of $11. How many notepads did she buy?
2. For a field trip 28 students rode in cars and the rest filled six buses. How many students were in each bus if 232 students were on the trip?
3. Joe’s bikes rents bikes for $15 plus $8 per hour. Julia paid $47 to rent a bike. For how many hours did she rent the bike?
4. Shelly had $24 to spend on five pencils. After buying them, she is left with $14. How much did each pencil cost?
5. Ted sold half of his comic books and then bought 15 more. He now has 45. With how many did he begin?

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