Mastering Word Problems in Linear Equations

Table of Contents

1. Introduction
2. Problem 1: Cost of Bottled Water and Candy Bar
   1. Step 1: Understanding the Information
   2. Step 2: Identifying Variables
   3. Step 3: Writing Equations
   4. Step 4: Solving the Equations
   5. Step 5: Putting the Solution into Context
3. Problem 2: Number of Paperback and Hardcover Books
   1. Step 1: Understanding the Problem
   2. Step 2: Identifying Variables
   3. Step 3: Writing Equations
   4. Step 4: Solving the Equations
   5. Step 5: Putting the Solution into Context
4. Conclusion

Problem 1: Cost of Bottled Water and Candy Bar

In this problem, We Are given the information that You purchased two bottled waters and three candy bars for $5.25, while your friend purchased five bottled waters and one candy bar for $8.25. The task is to determine the cost of a bottled water and the cost of a candy bar.

Step 1: Understanding the Information

To begin, let's Read through the problem and identify the important details. We are given two sets of purchases: yours and your friend's. We know the quantity of each item for both sets of purchases and the total amount spent.

Step 2: Identifying Variables

Next, let's identify the variables we need to find. In this case, we are looking for the cost of a bottled water (B) and the cost of a candy bar (C).

Step 3: Writing Equations

Based on the information provided, we can write two equations to represent the purchases made. The first equation represents your purchases:

2B + 3C = 5.25

The Second equation represents your friend's purchases:

5B + C = 8.25

Step 4: Solving the Equations

To solve this problem, we will use the elimination method. We will multiply the second equation by -3 and add it to the first equation to eliminate the variable C.

-3(5B + C) = -3(8.25) -15B - 3C = -24.75

Adding the two equations together, we get:

2B + 3C + (-15B - 3C) = 5.25 + (-24.75) -13B = -19.50

Dividing both sides by -13, we find that B is equal to 1.50.

Substituting this value back into one of the original equations, we can solve for C:

2(1.50) + 3C = 5.25 3 + 3C = 5.25 3C = 2.25 C = 0.75

Therefore, a bottled water costs $1.50 and a candy bar costs $0.75.

Step 5: Putting the Solution into Context

In conclusion, based on the given purchases and total amounts spent, we have determined that a bottled water costs $1.50 and a candy bar costs $0.75.

Problem 2: Number of Paperback and Hardcover Books

In this problem, you purchased a total of ten books, some of which are paperbacks and some are hardcovers. The number of paperbacks is two less than three times the number of hardcover books. We need to find out how many books of each Type you bought.

Step 1: Understanding the Problem

Let's read through the problem and gain an understanding of what it is asking. The problem states that the total number of books purchased is ten, and it provides information about the relationship between the number of paperbacks and hardcovers.

Step 2: Identifying Variables

In this case, we need to find the number of paperbacks (p) and the number of hardcover books (h).

Step 3: Writing Equations

Using the given information, we can write the following equation:

p + h = 10

And based on the relationship between paperbacks and hardcovers, we can write:

p = 3h - 2

Step 4: Solving the Equations

To solve this problem, we will use the substitution method. We will substitute the expression for p from the second equation into the first equation, and solve for h.

Substituting 3h - 2 for p, we have:

3h - 2 + h = 10 4h - 2 = 10 4h = 12 h = 3

Therefore, you purchased 3 hardcover books.

Now, substituting h = 3 into the equation p = 3h - 2, we can solve for p:

p = 3 * 3 - 2 p = 9 - 2 p = 7

Hence, you bought 7 paperback books.

Step 5: Putting the Solution into Context

In conclusion, based on the given information, you purchased 3 hardcover books and 7 paperback books.

Conclusion

In this article, we discussed two problems involving purchases and quantities. We solved both problems step by step, using the appropriate mathematical methods. By following these steps, we were able to determine the cost of a bottled water and a candy bar, as well as the number of paperback and hardcover books purchased. These problem-solving skills can be applied to various real-life situations, helping us make informed decisions and solve everyday challenges efficiently and effectively.