Problem 60 - College Algebra

Question

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule. At a market, the three most popular vegetables make up 53\(\%\) of vegetable sales. Corn has 4\(\%\) higher sales than broccoli, which has 5\(\%\) more sales than onions. What percentage does each vegetable have in the market share?

Step-by-Step Solution

Answer

Onions: 13\%, Broccoli: 18\%, Corn: 22\%.

Step 1: Define Variables

Let's define variables for the market share percentages of each vegetable. Let \( x \) be the percentage of the market that onions have, \( y \) for broccoli, and \( z \) for corn.

Step 2: Set Up Equations

Based on the problem statement, we can create the following equations:1. The sum of the percentages is given by: \[ x + y + z = 53 \]2. Corn has a 4\% higher market share than broccoli: \[ z = y + 4 \]3. Broccoli has 5\% more sales than onions: \[ y = x + 5 \]

Step 3: Substitute and Simplify Equations

Substitute the expressions for \( z \) and \( y \) into the first equation to get one equation with a single variable:- Substitute for \( z \): \[ x + y + (y + 4) = 53 \]- Simplify: \[ x + 2y + 4 = 53 \] \[ x + 2y = 49 \]

Step 4: Further Substitution

Substitute \( y = x + 5 \) into the equation \( x + 2y = 49 \):- Expand: \[ x + 2(x + 5) = 49 \] \[ x + 2x + 10 = 49 \]- Simplify: \[ 3x + 10 = 49 \] \[ 3x = 39 \]

Step 5: Solve for Onion Market Share \(x\)

Solve for \( x \):\[ x = \frac{39}{3} \]\[ x = 13 \]

Step 6: Solve for Broccoli Market Share \(y\)

Now that we have \( x = 13 \), substitute into the equation for \( y \):\[ y = x + 5 \]\[ y = 13 + 5 \]\[ y = 18 \]

Step 7: Solve for Corn Market Share \(z\)

Lastly, use the value of \( y = 18 \) to find \( z \):\[ z = y + 4 \]\[ z = 18 + 4 \]\[ z = 22 \]

Step 8: Verify the Solution

Verify that the sum matches the given equation:\[ x + y + z = 13 + 18 + 22 = 53 \]The sum matches the original condition, verifying our solution is correct.

Key Concepts

System of Linear Equations

Understanding how to create and solve a system of linear equations is vital in interpreting problems like the one here. A system of linear equations consists of two or more linear equations involving the same set of variables. Here, the task is to express the behavior of market shares with mathematical equations.
To start, we assign variables: let \( x \) be the percentage of the market that onions have, \( y \) for broccoli, and \( z \) for corn. From the problem statement, we derive three equations:

• The sum of the market shares for onions, broccoli, and corn is 53\(%\): \[ x + y + z = 53 \]
• Corn has a 4\(%\) higher market share than broccoli: \[ z = y + 4 \]
• Broccoli has a 5\(%\) higher market share than onions: \[ y = x + 5 \]

These equations form the system of linear equations we need to solve to find the market share of each vegetable.

Market Share Percentage

Market share percentage is a concept used to understand the portion of a market controlled by a company, product, or—in this case—vegetables. Here, the vegetables in question are onions, broccoli, and corn. Understanding market share involves comparing how much of the total market each item commands.

• Onions have a percentage represented by \( x \).
• Broccoli's share is represented by \( y \), and it has a little more market share than onions.
• Corn, represented by \( z \), tops the list with a slightly higher percentage than broccoli.

These comparisons are turned into our system of equations, as they help refine the relative weights each vegetable has in the total market share of 53\(%\). Correctly calculating each value provides a clearer picture of their impact in the marketplace.

Substitution Method

The substitution method is a common strategy for solving systems of linear equations. It involves solving one of the equations for one variable and then substituting that expression into the other equations. This problem illustrates how substitution can simplify solving for unknowns.
We start with our three equations:

• The total percentage: \[ x + y + z = 53 \]
• Corn's share over broccoli: \[ z = y + 4 \]
• Broccoli's share over onions: \[ y = x + 5 \]

By substituting in the relationships involving \( y \) and \( z \) into the first equation, we simplify it to one variable.
This approach reduces the complexity, letting us solve for \( x \), and consequently \( y \) and \( z \). With \( x = 13 \), \( y = 18 \), and \( z = 22 \), we find each vegetable's market share and validate the solution by ensuring they sum up to the known total market share of 53\(%\). The substitution method can be incredibly powerful when strategically applied to interrelated equations.