Problem 66 - Precalculus Mathematics for Calculus

Question

A customer in a coffee shop purchases a blend of two coffees: Kenyan, costing \(\$ 3.50\) a pound, and Sri Lankan, costing \(\$ 5.60\) a pound. He buys 3 Ib of the blend, which costs him \(\$ 11.55 .\) How many pounds of each kind went into the mixture?

Step-by-Step Solution

Answer

2.5 pounds of Kenyan coffee and 0.5 pounds of Sri Lankan coffee.

Step 1: Define Variables

Let \( x \) represent the pounds of Kenyan coffee and \( y \) represent the pounds of Sri Lankan coffee. We are asked to find the values of \( x \) and \( y \).

Step 2: Set Up the Equations

Using the information given, set up two equations. The first equation is the total weight: \( x + y = 3 \). The second equation involves the cost: \( 3.50x + 5.60y = 11.55 \).

Step 3: Solve the Total Weight Equation

From the equation \( x + y = 3 \), isolate \( y \) to get \( y = 3 - x \).

Step 4: Substitute and Solve the Cost Equation

Substitute \( y = 3 - x \) into the cost equation: \( 3.50x + 5.60(3 - x) = 11.55 \). Expand and simplify to solve for \( x \).

Step 5: Simplification Process

Simplify the equation: \( 3.50x + 16.80 - 5.60x = 11.55 \). Combine like terms: \( -2.10x + 16.80 = 11.55 \).

Step 6: Isolate the Variable

Move constants to the right: \( -2.10x = 11.55 - 16.80 \), which simplifies to \( -2.10x = -5.25 \).

Step 7: Solve for x

Divide both sides by \(-2.10\) to find \( x \): \( x = \frac{-5.25}{-2.10} \). Calculating gives \( x = 2.5 \).

Step 8: Solve for y

Substitute \( x = 2.5 \) back into \( y = 3 - x \) to find \( y \): \( y = 3 - 2.5 = 0.5 \).

Step 9: Verify the Solution

Check the solution by substituting \( x = 2.5 \) and \( y = 0.5 \) back into both original equations to ensure they are satisfied. Both equations hold true, confirming the solution is correct.

Key Concepts

System of Equations

In mathematics, a **system of equations** refers to a set of two or more equations that have common variables. The goal is to find the values of these variables that satisfy all equations simultaneously. In the presented problem, buying a coffee blend, we need to figure out how many pounds of Kenyan and Sri Lankan coffee were purchased. For this, two primary equations are set up:

• The **total weight equation**: This equation simply adds up the pounds of Kenyan and Sri Lankan coffee to equal the total weight of the mixture.
• The **cost equation**: This involves accounting for the different prices per pound of each coffee and equating the total to the amount spent on the blend.

These equations form a system that represents the entire situation. Solving them provides the amount of each coffee bought.

Substitution Method

The **substitution method** is one effective way to solve a system of equations. The idea is to solve one of the equations for a single variable and then substitute that expression into the other equation. Let's break it down based on the coffee blend problem:

• Firstly, you solve the total weight equation for one variable. In this problem, we solved for \( y \) as: \( y = 3 - x \).
• Next, you substitute this expression for \( y \) into the second equation (the cost equation). This converts the cost equation into one involving only \( x \).

This method reduces the system into a single variable equation, which is often much simpler to solve. Once you find one variable, you substitute it back to find the other.

Problem Solving Steps

Let's go over the specific **problem-solving steps** used in the coffee blend scenario. Following these steps can make working through similar systems clear and straightforward:1. **Define the variables**: Clearly state what each variable represents. Here, \( x \) and \( y \) represent the pounds of Kenyan and Sri Lankan coffee, respectively.2. **Set up equations** based on the problem statement. Translate the wording into math: the total weight and cost equations.3. **Solve the simpler equation**: The total weight equation was simpler. Solving it for one variable helps reduce complexity in the next step.4. **Substitute and tackle the more complex equation**: Insert the simplified expression in place of the variable in the cost equation, allowing for an easy path to find the primary unknown.5. **Perform simplification and isolation**: Use algebra to simplify and isolate variables. Work step by step, ensuring to maintain accuracy.6. **Solve for the remaining variable**: After finding \( x \), use it to determine \( y \).7. **Verify your results**: Substitute your solutions back into both original equations. Make sure both equations hold true to confirm the correctness of the solution. By adhering to these structured steps, you'll not only find the correct solution but also develop a clearer understanding of methodically solving systems of equations.