Problem 85 - Introductory and Intermediate Algebra for College Students 4th

Question

At the beginning of a semester, a student purchased eight pens and six pads for a total cost of 14.50 dollars . 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 dollars each, find the cost of one pen.

Step-by-Step Solution

Answer

a. The equation that reflects the given conditions would be \(8x + 6y = 14.50\). \nb. Upon substitution of the given pad cost and calculation, the cost of one pen comes out to be \(x = (14.50 - 6(0.75))/8\) dollars.

Step 1: Part a: Formulating the Equation

For eight pens and six pads, the student spent 14.50 dollars. One can express this relationship with a linear equation. Given the information above, particularly that \(x\) represents the cost of one pen and \(y\) represents the cost of one pad, the equation will be: \(8x + 6y = 14.50\)

Step 2: Part b: Calculation of the Cost of One Pen

The cost of one pad (\(y\)) is stated to be 0.75 dollars. This value can be substituted into the equation. Therefore, the equation becomes \(8x + 6(0.75) = 14.50\), and this simplifies to \(8x = 14.50 - 6(0.75)\). Solving for \(x\) or the cost of one pen, one gets \(x = (14.50 - 6(0.75))/8\).

Key Concepts

Variables in Equations

When dealing with linear equations, variables are essentially placeholders for unknown values that we aim to find.
In our exercise, there are two crucial variables:

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

By using variables, we simplify complex problems into manageable equations, allowing unknown quantities like the cost of a single pen to be determined through calculation.
Equations with more than one variable can appear daunting, but they provide a framework to insert known values and solve for what is missing.
Here, variables transform the practical problem of buying pens and pads into an algebraic expression.

Solving Equations

Solving equations is all about finding the unknown values of the given variables.
The central equation from the problem is given as \(8x + 6y = 14.50\).
This represents the total cost of eight pens and six pads. Initially, it might appear complex because of two unknowns: \(x\) and \(y\). First, we simplify by substituting known values.
The problem tells us the cost of one pad (\(y\)) is 0.75 dollars. So, the equation \(8x + 6(0.75) = 14.50\) uses that specific value.
Substitute and simplify: \(8x + 4.50 = 14.50\).
Isolate \(x\) by subtracting 4.50 from both sides, resulting in \(8x = 10.00\).Solve for \(x\) by dividing both sides by 8, leading to \(x = 1.25\). Thus, solving equations transitions abstract algebra into tangible meanings, such as discovering the cost of one pen is 1.25 dollars.

Cost Calculation

Cost calculation in this context involves understanding how to distribute expenses across the items purchased.
Knowing the total expenditure and the individual cost of some items helps us deduce the unknown costs.In the exercise, we know the total cost of pens and pads is 14.50 dollars.
By determining the cost of pads first (given as 0.75 dollars each), it becomes possible to calculate how much remains for the pens.

• Total known cost of pads: \(6 \times 0.75 = 4.50\) dollars.
• Remaining budget for pens: \(14.50 - 4.50 = 10.00\) dollars.
• Since eight pens need to fit this budget, divide to find each pen's cost: \(10.00/8\) = 1.25 dollars.

This systematic breakdown clarifies that cost calculation is not just about mathematics but also about logical reasoning, making sure every cent is accounted for in purchasing decisions.