Problem 64 - Precalculus: Mathematics for Calculus

Question

Fruit Stand \(A\) fruit stand sells two varieties of strawberries: standard and deluxe. A box of standard strawberries sells for \(\$ 7,\) and a box of deluxe strawberries sells for \(\$ 10 .\) In one day the stand sold 135 boxes of strawberries for a total of \(\$ 1110\). How many boxes of each type were sold?

Step-by-Step Solution

Answer

80 boxes of standard strawberries and 55 boxes of deluxe strawberries were sold.

Step 1: Set Up Variables

Let \( x \) represent the number of standard boxes sold, and \( y \) represent the number of deluxe boxes sold. We are given two types of information: the total number of boxes sold and the total revenue.

Step 2: Create Equations

First, create an equation for the total number of boxes sold: \[ x + y = 135 \] Next, create an equation for the total revenue: \[ 7x + 10y = 1110 \]

Step 3: Solve for One Variable

We will solve the first equation for \( x \). The equation is: \[ x + y = 135 \] Subtract \( y \) from both sides to get \[ x = 135 - y \]

Step 4: Substitute and Solve

Substitute \( x = 135 - y \) into the second equation: \[ 7(135 - y) + 10y = 1110 \] Expanding the equation gives: \[ 945 - 7y + 10y = 1110 \] Combine like terms to get: \[ 3y = 1110 - 945 \] Solve for \( y \) : \[ 3y = 165 \]\[ y = 55 \]

Step 5: Find the Other Variable

With \( y = 55 \), plug back into \( x = 135 - y \): \[ x = 135 - 55 \] \[ x = 80 \]

Step 6: Verify the Solution

Verify the solution is correct by substituting \( x = 80 \) and \( y = 55 \) back into both original equations:1. \( 80 + 55 = 135 \) - correct.2. \( 7(80) + 10(55) = 560 + 550 = 1110 \) - correct.Both checks confirm the solution is valid.

Key Concepts

Variable Substitution

Variable substitution is a powerful tool when working with systems of linear equations. This process allows us to solve one equation for a specific variable, and then "substitute" this expression into another equation. By doing this, we can reduce the number of variables in the equation and solve it step by step.

Let's break this down with an example from our fruit stand problem. We initially set up two equations: one for the total number of boxes and the other for total revenue. To simplify the equations, we solve the first equation for one of the variables, say, \( x \) or \( y \). In this case, we chose \( x \), resulting in the equation \( x = 135 - y \).

Substitution involves replacing \( x \) in the second equation (which involves revenues) with \( 135 - y \), leaving us with a single variable \( y \) to solve for. This makes the equation much simpler and brings us closer to solving the problem.

Equation Setup

Setting up equations correctly is crucial for solving any systems of linear equations problem. This step lays the foundation for the entire solution.

In our original problem involving standard and deluxe strawberries, we start by identifying the core pieces of information:

• The total number of boxes sold: 135 (Equation: \( x + y = 135 \))
• The total revenue from these boxes: \$ 1110 (Equation: \( 7x + 10y = 1110 \))

Setting up the equations involves translating these pieces of information into mathematical expressions. For example, the equation \( x + y = 135 \) is derived from the total number of boxes, while \( 7x + 10y = 1110 \) correlates with the revenue.

This equation setup is vital because it directly reflects the relationship described in the problem, providing the basis for all subsequent steps.

Problem Solving using Algebra

Solving problems using algebra involves strategic steps and a methodologic approach. Starting with substitution, after breaking down each equation and rewriting it, we can solve for one variable at a time.

In our example, after substituting \( x = 135 - y \) into the revenue equation \( 7x + 10y = 1110 \), we expanded and simplified the equation to get \( 3y = 165 \). Solving that yields \( y = 55 \).

Finding \( y \) allowed us to plug back into the equation \( x = 135 - y \) to find \( x = 80 \). With both variables found, the last step is always to verify by substituting back into the original equations to ensure consistency.

By solving step by step, and verifying the results, algebra provides a solid framework to approach and solve complex problems effectively. This process is not just mechanical but involves critical thinking and logical reasoning to ensure the final answers are correct.