Problem 23 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A bookstore buys 35 books for \(\$ 271 .\) Some of the books cost \(\$ 7\) each and the remainder cost \(\$ 9\) each. How many of each type were bought?

Step-by-Step Solution

Answer

22 books cost \( \$7 \) each and 13 books cost \( \$9 \) each.

Step 1: Define Variables

Let \( x \) represent the number of \( \$7 \) books, and \( y \) represent the number of \( \$9 \) books.

Step 2: Set Up Equations

The total number of books is given by the equation: \[ x + y = 35 \]The total cost equation is given by: \[ 7x + 9y = 271 \].

Step 3: Solve for One Variable

From the first equation, solve for \( y \):\[ y = 35 - x \]

Step 4: Substitute and Solve

Substitute \( y = 35 - x \) into the second equation: \[ 7x + 9(35 - x) = 271 \].Simplifying, we get: \[ 7x + 315 - 9x = 271 \]\[ -2x + 315 = 271 \]\[ -2x = -44 \]\[ x = 22 \]

Step 5: Find the Other Variable

Substitute \( x = 22 \) back into the equation \( y = 35 - x \): \[ y = 35 - 22 \]\[ y = 13 \]

Step 6: State the Solution

The bookstore bought 22 books that cost \( \$7 \) each and 13 books that cost \( \$9 \) each.

Key Concepts

Solving Equations

Solving equations is one of the core concepts in algebra that allows us to find the unknown values given certain conditions. In this example, we were given two different sets of books with specific costs and a total amount spent. By setting up equations, we can methodically solve for each variable.

When solving equations, we often start by simplifying each equation, isolating one variable, and then using that information to find the value of other variables. It's like unraveling a mystery! First, break the problem into smaller, manageable parts and then bring it all together.

Defining Variables

Defining variables is a critical step in solving word problems. It helps convert a real-world problem into a mathematical formulation. In the bookstore problem, we defined:

• \( x \) as the number of \$7 books
• \( y \) as the number of \$9 books

By doing so, we create a clear representation of what we're trying to find.

Properly defining variables ensures that each step of the solution process has a point of reference. This clarity makes it easier to follow through the computations and verify if the results make sense. Always keep clear definitions to avoid confusion.

Substitution Method

The substitution method is an effective algebraic technique to solve systems of equations. After defining our variables and setting up equations: \[ x + y = 35 \]\[ 7x + 9y = 271 \]

We solved for \( y \) from the first equation and substituted this value into the second equation:
\[ y = 35 - x \]
\[ 7x + 9(35 - x) = 271 \]

This substitution reduces the problem to an equation with one variable, making it easier to solve. Once we found \( x \), we substituted back to get \( y \). The substitution method simplifies the complex process of solving multiple variables into manageable steps.

System of Linear Equations

As shown in our example, a system of linear equations is a collection of one or more linear equations involving the same set of variables. Here, the linear equations were: \[ x + y = 35 \]\[ 7x + 9y = 271 \]

Solving a system of linear equations involves finding a common solution that satisfies all the given equations simultaneously. Different methods to solve these include substitution, elimination, or graphical representation.

It’s pivotal to understand that a solution to a system of equations represents the point of intersection of the lines (or planes in higher dimensions) described by each equation. Mastering this concept will greatly enhance your problem-solving toolkit for algebraic word problems.