Problem 68 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system for \(x, y,\) and \(z\). The top three oil producers in the United States in a certain year are the Gulf of Mexico, Texas, and Alaska. The three regions were responsible for \(64 \%\) of the United States oil production. The Gulf of Mexico and Texas combined for \(47 \%\) of oil production. Texas produced 3\% more than Alaska. What percent of United States oil production came from these regions?

Step-by-Step Solution

Answer

Gulf of Mexico: 27%, Texas: 20%, Alaska: 17%.

Step 1: Define Variables

Let's define the variables that represent the percent oil production of each region. Let \( x \) be the percentage of oil production from the Gulf of Mexico, \( y \) be the percentage from Texas, and \( z \) be the percentage from Alaska.

Step 2: Set Up the Equations

From the problem, we have the following equations:1. \( x + y + z = 64 \) (total oil production from the three regions is 64%)2. \( x + y = 47 \) (Gulf of Mexico and Texas combined for 47% of production)3. \( y = z + 3 \) (Texas produced 3% more than Alaska)

Step 3: Solve the System of Equations

First, use Equation 2 to solve for \( x \):\[ x = 47 - y \]Next, substitute \( y = z + 3 \) from Equation 3 into Equation 1: \[ x + (z + 3) + z = 64 \]which simplifies to:\[ x + 2z + 3 = 64 \]\[ x + 2z = 61 \]Now, substitute \( x = 47 - y = 47 - (z + 3) = 44 - z \) into \( x + 2z = 61 \):\[ 44 - z + 2z = 61 \]\[ z = 17 \]

Step 4: Find Remaining Values

Use the value of \( z \) to find \( y \) and \( x \):From Equation 3, \( y = z + 3 = 17 + 3 = 20 \)From \( x = 47 - y = 47 - 20 = 27 \)

Step 5: Verify the Solution

Plug \( x = 27 \), \( y = 20 \), and \( z = 17 \) back into the original equations to verify they hold true:1. \( x + y + z = 27 + 20 + 17 = 64 \)2. \( x + y = 27 + 20 = 47 \)3. \( y = z + 3 = 17 + 3 = 20 \)The solution is verified as correct.

Key Concepts

Variables Definition

In algebra, especially when dealing with systems of linear equations, defining variables is the first crucial step. Here, we are tasked to determine the percentage of oil production from three key regions: the Gulf of Mexico, Texas, and Alaska. For simplicity and clarity, we assign:

• \( x \) as the percentage of production from the Gulf of Mexico,
• \( y \) as the percentage from Texas,
• \( z \) as the percentage from Alaska.

Variables serve as placeholders for unknown values and are essential for setting up equations later. Establishing these definitions allows us to translate the real-world problem into a mathematical model, making it easier to work with and analyze.

Equation Solving

Once we've defined the variables, the next step is to create equations based on the problem statement. In our exercise, three key relationships were identified:

• The total oil production from the three regions equals 64%, giving us the equation \( x + y + z = 64 \).
• The Gulf of Mexico and Texas together account for 47% of the production, translating into \( x + y = 47 \).
• Texas produces 3% more than Alaska, leading to \( y = z + 3 \).

These equations summarize the conditions given in the problem. Solving them involves logical steps, simplified calculations, and substitution, which transfer the real-world scenario into simplified mathematical language.

Substitution Method

The substitution method is a powerful tool for solving systems of equations. It requires expressing one variable in terms of another, then substituting it into the other equations to simplify the solving process.

• From equation \( x + y = 47 \), we resolve one of the variables in terms of the other: \( x = 47 - y \).
• Next, incorporate \( y = z + 3 \) from the third equation into the other:
  • \( x + (z + 3) + z = 64 \) simplifies to \( x + 2z = 61 \).
  • Substitute \( x = 44 - z \) from earlier into this simplified equation.

This method reduces the complexity of solving multiple equations, allowing you to find each variable sequentially.

Verification of Solution

After determining potential solutions for the variables, it's essential to verify their correctness. Re-evaluating ensures your values satisfy all of the original equations:

• Check \( x + y + z = 64 \): Plugging in \( x = 27 \), \( y = 20 \), and \( z = 17 \) gives us \( 27 + 20 + 17 = 64 \).
• Validate \( x + y = 47 \): The sum \( 27 + 20 \) indeed equals 47.
• Confirm \( y = z + 3 \): Observing \( 20 = 17 + 3 \) verifies this is accurate.

Verification helps ensure that calculations are correct and solutions are consistent, reducing the chances of error and increasing confidence in the result.