Problem 114 - College Algebra with Modeling and Visualization

Question

In 2006 , the commercial sector used 3.74 quadrillion (10") Btu of energy from natural gas and petroleum. It used 3.02 quadrillion Btu more natural gas than petroleum. (a) Write a system of equations whose solution gives the consumption of natural gas and petroleum (in quadrillion Bta). (b) Solve the system of equations. (c) Is your system consistent or inconsistent? If it is consistent, state whether the equations are dependent or independent.

Step-by-Step Solution

Answer

Natural gas: 3.38 quadrillion Btu, Petroleum: 0.36 quadrillion Btu; consistent and independent system.

Step 1: Define Variables

Let \( x \) represent the energy consumption from natural gas (in quadrillion Btu) and \( y \) represent the energy consumption from petroleum (in quadrillion Btu).

Step 2: Write the Equations

According to the problem, the total consumption is:\[ x + y = 3.74 \]Additionally, it states that natural gas usage exceeds petroleum by 3.02 quadrillion Btu:\[ x = y + 3.02 \]

Step 3: Substitute One Equation into the Other

Substitute \( x = y + 3.02 \) into \( x + y = 3.74 \):\[ (y + 3.02) + y = 3.74 \]

Step 4: Solve for y

Simplify the equation:\[ 2y + 3.02 = 3.74 \]Subtract 3.02 from both sides:\[ 2y = 0.72 \]Divide by 2:\[ y = 0.36 \]

Step 5: Solve for x

Substitute \( y = 0.36 \) back into \( x = y + 3.02 \):\[ x = 0.36 + 3.02 \]\[ x = 3.38 \]

Step 6: Check Consistency and Dependency

Since both equations intersect at the point \((x, y) = (3.38, 0.36)\), the system is consistent and independent.

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. When dealing with systems of equations, your goal is to find common values for the variables that satisfy all the equations simultaneously. In this exercise, we are tasked with identifying how natural gas and petroleum energy consumption add up using two equations. First, we introduce variables: let \( x \) represent the energy consumed from natural gas and \( y \) represent the energy from petroleum. The problem gives us two key pieces of information:

• The total energy consumption from both sources is 3.74 quadrillion Btu.
• Natural gas consumption is 3.02 quadrillion Btu more than petroleum consumption.

Using this information, we write:

• Equation 1: \( x + y = 3.74 \)
• Equation 2: \( x = y + 3.02 \)

These two equations describe our system. To solve them, we can use a variety of methods such as substitution or elimination. In this example, we substituted the second equation into the first to find the values of \( x \) and \( y \). Systems of equations are foundational in algebra and occur frequently in real-world problem-solving scenarios.

Energy Consumption

Energy consumption is a critical aspect when analyzing resource usage, particularly in commercial sectors. Here, we examine the energy consumed by natural gas and petroleum, quantified in quadrillion British thermal units (Btu), a standard unit of measurement in energy. The exercise illustrates a common real-world scenario: determining how different energy sources contribute to total consumption. The total energy consumed by these two resources is given as 3.74 quadrillion Btu. It's essential to understand each source's contribution, not just the total. You might want to consider:

• Changes over time: As energy needs fluctuate, so does the consumption of each source.
• Comparative analysis: With the given data, you can compare energy usage patterns.

In this scenario, natural gas is highlighted as being consumed at a significantly higher rate than petroleum by 3.02 quadrillion Btu. Understanding energy consumption helps in resource allocation and planning for sustainable usage in commercial sectors.

Problem Solving Steps

Breaking down a problem into clear, manageable steps is crucial for efficient problem-solving, particularly with systems of equations. Let's go through the key steps in this exercise to unlock the solution: **1. Define A Logical Framework**When starting, we defined variables where \( x \) represented natural gas and \( y \) represented petroleum. Defining variables is the cornerstone of formulating equations effectively. **2. Formulate Equations Based on the Problem Description**The problem's details guide you in forming equations. We have:

• Total consumption equation: \( x + y = 3.74 \)
• Relative consumption equation: \( x = y + 3.02 \)

**3. Substitute and Simplify**Using substitution, one equation can be inserted into another, making it simpler to find the value of one variable. Here, substituting \( x = y + 3.02 \) into \( x + y = 3.74 \) results in:\( 2y + 3.02 = 3.74 \). Solving then gives \( y = 0.36 \).**4. Solve for Remaining Variables**With one variable solved, substitute \( y = 0.36 \) back to solve for \( x \): \( x = 3.38 \).**5. Validate the Solution**Finally, verifying that your solutions satisfy the original equations ensures their correctness. Checking consistency and dependency ensures we're dealing with a solvable system.By utilizing these problem-solving steps, one can methodically tackle complex mathematical problems with ease and accuracy.