Problem 25 - Algebra A Combined Function

Question

Solve. For each exercise, a table is given for you to complete and use to write an equation that models the situation. Community Coffee Company wants a new flavor of Cajun coffee. How many pounds of coffee worth \(\$ 7\) a pound should be added to 14 pounds of coffee worth \(\$ 4\) a pound to get a mixture worth \(\$ 5\) a pound?

Step-by-Step Solution

Answer

Add 7 pounds of coffee worth \$7 per pound.

Step 1: Define Variables

Let the number of pounds of coffee worth $7 per pound be denoted as \( x \).

Step 2: Set Up Equation for Total Weight

The total weight of the coffee mixture will be the sum of the 14 pounds of coffee worth \(4 per pound and the \( x \) pounds of coffee worth \)7 per pound. Hence, the equation for total weight is \( 14 + x \).

Step 3: Set Up Equation for Total Cost

The total cost of the mixture can be expressed in terms of the individual weights and prices. The coffee worth \(4 costs \( 14 \times 4 \) dollars, and the coffee worth \)7 costs \( 7x \) dollars, giving us a total cost of \( 14 \times 4 + 7x \).

Step 4: Equation for Desired Price

We need the mixture to cost $5 per pound at the end. Therefore, \( 5(14 + x) \) equals the total cost, given by \( 14 \times 4 + 7x \). Hence, the equation is \( 5(14 + x) = 14 \times 4 + 7x \).

Step 5: Solve the Equation

Expand and simplify the linear equation: \[ 5(14 + x) = 56 + 7x \] which simplifies to \[ 70 + 5x = 56 + 7x \]. Gather all \( x \)-terms on one side: \[ 70 - 56 = 7x - 5x \], giving \[ 14 = 2x \]. Divide by 2 to solve for \( x \): \( x = 7 \).

Step 6: Verify the Solution

Plug \( x = 7 \) back into the total equations to verify. Total weight: \( 14 + 7 = 21 \) pounds. Total cost: \( 14 \times 4 + 7 \times 7 = 56 + 49 = 105 \). Price per pound: \( 105 / 21 = 5 \), which matches the desired \$5 per pound.

Key Concepts

Linear Equations

Linear equations are the backbone of algebraic problem-solving, especially when we're dealing with mixture problems, like in our given exercise. These equations involve variables raised to the power of one and typically look like this:

• "ax + b = c",
• where "a," "b," and "c" are constants,
• and "x" is the variable we solve for.

In our Cajun coffee problem, the main equation \[ 5(14 + x) = 14 \times 4 + 7x \] represents a linear equation. Here, everything balances around the variable "x," which stands for the unknown pounds of $7 coffee to be used in the mixture.

Solving linear equations usually involves simple operations such as:

• Expanding terms,
• Rearranging components,
• And isolating the variable.

These actions allow us to pinpoint the exact value of "x". In the context of mixtures, this enables us to see how much of each component we need to reach a certain target.

Mixture Problems

Mixture problems are practical applications of linear equations. They involve combining substances at different variables – usually different prices or concentrations – to create a desired final result. In our specific coffee problem, we're looking to blend:

• 14 pounds of \(4 per pound coffee,
• with an unknown quantity of \)7 per pound coffee,
• to achieve a blend worth \(5 per pound.

Such problems typically require setting up and solving an equation that balances the total cost of components against the total cost of the desired mixture.

The steps include:

• Defining variables for unknown quantities,
• Creating equations based on total weight, total cost, and the targeted price per unit.

In our exercise, the equation was expertly linked to ensure that the blend weight "14 + x" and the total cost "14 \times 4 + 7x" support the final product's value of \)5 per pound, translating an abstract problem into concrete steps.

Problem-Solving Steps

Methodically solving algebraic problems begins with structured problem-solving steps. This systematic approach was followed in solving the coffee mixture problem. Here's a breakdown:

• **Step 1:** Define Variables: Identify what you need to find, labeling it with a variable like "x."
• **Step 2:** Set Up Equations: Develop equations reflecting the relationship between variables (weight, cost) and constants (price per unit).
• **Step 3:** Solve and Verify: Solve these equations, ensuring all units and relationships make sense.

Verification is essential. It confirms the solution matches real-world expectations. For instance, after calculating "x = 7," verifying involves checking that integrating 7 pounds of $7 coffee leads to a blend worth exactly $5 per pound.

These steps are crucial for dissecting and understanding mixture problems or any algebraic equation. They guide you from unknowns to a verified solution, enhancing understanding along the journey.