Problem 85 - Introductory Algebra for College Students

Question

At the beginning of a semester, a student purchased cight pens and six pads for a total cost of \(\$ 14.50 .\) a. If \(x\) represents the cost of one pen and \(y\) represents the cost of one pad, write an equation in two variables that reflects the given conditions. b. If pads cost \(\$ 0.75\) each, find the cost of one pen.

Step-by-Step Solution

Answer

The cost of one pen is $1.25.

Step 1: Write the First Equation

Translate the first problem statement into an equation. The total cost of eight pens and six pads is $14.50. Therefore, using \(x\) as the cost of one pen and \(y\) as the cost of one pad, this can be written as the equation: \(8x + 6y = 14.50\)

Step 2: Substitute the Value of y into the Equation

The second part of the problem provides that the cost of a pad (\(y\)) is $0.75. Substitute \(y=0.75\) into the original equation, which gives: \(8x + 6*(0.75) = 14.50\). Simplify this to get: \(8x + 4.5 = 14.50\)

Step 3: Solve for x

To isolate \(x\), subtract 4.5 from both sides of the equation: \(8x = 14.50 - 4.5\), then divide both sides by 8 to get: \(x = 1.25\). Therefore, the cost of one pen is $1.25.

Key Concepts

Linear Equations

Linear equations are equations of the first degree, which means the variables are raised only to the power of one. In this exercise, we deal with a linear equation that helps us figure out the costs of pens and pads. An example of a linear equation based on the problem is:

• \(8x + 6y = 14.50\)

This equation comes from a real-world scenario, where a student buys eight pens and six pads for $14.50. Here,

• \(x\) represents the cost of one pen
• \(y\) represents the cost of one pad

Linear equations like this are useful because they represent relationships where the change in one quantity is proportional to the change in another. This is foundational in algebra, assisting in determining unknown values.

Variables

Variables are symbols that represent unknown numbers or values in equations. In the context of this problem, we use the variables \(x\) and \(y\).

• \(x\) is the variable for the cost of one pen.
• \(y\) is the variable for the cost of one pad.

Variables are incredibly important because they allow us to set up equations based on real-life situations without knowing all the values involved from the beginning. Once defined, you can manipulate these symbols mathematically to discover unknown quantities. Using variables lets us create a general formula, which we can then adjust as we get more specific information, like the given cost of pads in part (b).

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves substituting the value of one variable into another equation.
In this exercise, we know the cost of a pad is $0.75, hence we substitute \(y = 0.75\) into the original equation:

• \(8x + 6*(0.75) = 14.50\)

This step simplifies the equation to something that contains only one variable, making it easier to solve for the unknown \(x\). Once you substitute and simplify, you can isolate the variable and perform basic arithmetic to find the solution. The substitution method is efficient because it reduces a system of equations into a single equation with one variable.

Cost Analysis

Cost analysis is the process of determining the financial outcome of buying goods, in this case, pens and pads. Through the linear equation \(8x + 6y = 14.50\), we evaluate how much each pen and pad contributes to the total cost.
Once we know \(y = 0.75\), the problem becomes a cost analysis exercise focusing on the pens. By substituting \(y\) in the equation, we're looking to uncover the hidden cost of each pen. This kind of problem is common in budgeting and shopping scenarios, helping people understand where and how their money is spent. Analyzing costs using algebraic equations enables better planning and financial decision-making. You see, once we found that the cost for one pen, \(x\), is $1.25, it completes our understanding of the total expense.