Problem 58 - Algebra A Combined Function

Question

For the years 1995 through \(2005,\) the annual percent \(y\) of U.S. households that used a wall or floor furnace to heat their houses is given by the equation \(y=-0.04 x+5.1,\) where \(x\) is the number of years after \(1995 .\) For the same period, the annual percent \(y\) of U.S. households that used fireplaces or wood stoves to heat their homes is given by \(y=-0.31 x+5.3\), where \(x\) is the number of years after 1995. (Source: U.S. Census Bureau, American Housing Survey Branch) a. Use the substitution method to solve this system of equations. $$ \left\\{\begin{array}{l} y=-0.04 x+5.1 \\ y=-0.31 x+5.3 \end{array}\right. $$ Round your answer to the nearest whole numbers. b. Explain the meaning of your answer to part (a). c. Sketch a graph of the system of equations. Write a sentence describing the use of wall furnaces or fireplaces or wood stoves for heating homes between 1995 and 2005 .

Step-by-Step Solution

Answer

In 1996, about 5% of households used both heating methods equally.

Step 1: Setting Equations Equal

Since both equations equal \(y\), you can set them equal to each other: \(-0.04x + 5.1 = -0.31x + 5.3.\)

Step 2: Simplifying the Equation

To simplify, first add \(0.31x\) to both sides: \(0.27x + 5.1 = 5.3.\)

Step 3: Isolating x

Subtract 5.1 from both sides to isolate the term with \(x\): \(0.27x = 0.2.\)

Step 4: Solving for x

Divide both sides by \(0.27\) to solve for \(x\): \(x = \frac{0.2}{0.27} \approx 0.74.\)

Step 5: Rounding the Result

Round \(x\) to the nearest whole number to find the year: \(x \approx 1.\)

Step 6: Interpreting the Result

Substitute \(x = 1\) back into one of the original equations to find \(y\): For \(y = -0.04x + 5.1\), \(y = -0.04(1) + 5.1 = 5.06.\) Round \(y\) to the nearest whole number.Thus, \(y \approx 5.\)

Step 7: Explaining the Meaning

When \(x = 1\), the two heating methods had the same percentage use, roughly 5%, meaning in 1996, both methods were used by an equal percentage of households.

Step 8: Sketching the Graph

For a graph, plot the two linear equations: 1. \(y = -0.04x + 5.1\) is a line with a gentle negative slope 2. \(y = -0.31x + 5.3\) is a line with a steeper negative slope. The point where they meet, \(x = 1\), \(y = 5\), shows when the usage percentages were equal.

Step 9: Describing Graph Outcome

Between 1995 and 2005, the percentage of U.S. households using wall or floor furnaces declined slightly, while the percentage using fireplaces or wood stoves decreased more rapidly. They were equal in 1996.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations. This method is particularly useful when you have an equation already solved for one variable, which allows you to easily substitute it into another equation.

Here's how the substitution method works in the context of our original exercise about heating methods in U.S. households:

• Start by setting both equations equal to the same variable. In this scenario, both equations are given in terms of the variable \(y\).
• Since \(y = -0.04x + 5.1\) and \(y = -0.31x + 5.3\), you equate them: \(-0.04x + 5.1 = -0.31x + 5.3\).
• From here, you simplify and solve for another variable, in this case, \(x\). This step requires rearranging the equation to isolate \(x\).

The substitution method is particularly advantageous because it allows you to find one variable first and then easily calculate the corresponding values for the other variables by substituting back. This way, you can determine the intersection point of the two linear equations represented by the system.

Linear Equations

Linear equations are mathematical expressions that depict a straight line when graphed. Each equation usually involves constant coefficients and one or more variables without any variables being raised to a power greater than one.

In the exercise, both given equations are linear, described as:

• \(y = -0.04x + 5.1\)
• \(y = -0.31x + 5.3\)

Each equation describes the relationship between the years after 1995 and the percentage of U.S. households using different heating methods during those years.

The coefficients of \(x\), \(-0.04\) and \(-0.31\), represent the rate at which each heating method's usage changed over time. Meanwhile, the constant terms, 5.1 and 5.3, depict the initial percentages in the year 1995. Understanding these components helps you interpret how the usage of these heating methods evolved over the years.

Graphical Interpretation

Graphing systems of linear equations provides a visual representation of their solutions, showing how they intersect on a coordinate plane. When graphing our two linear equations, you'll see:

• The line \(y = -0.04x + 5.1\) with a gentle negative slope, representing the slow decrease in usage of wall or floor furnaces.
• The line \(y = -0.31x + 5.3\) illustrating a steeper drop, indicating the quicker decline in the use of fireplaces or wood stoves.

Where these two lines intersect is particularly important. In this instance, they meet at the point \((x = 1, y = 5)\), which means in 1996, both heating methods were equally used by approximately 5% of U.S. households. This intersection signifies a moment of balance between the two trends.

Understanding how to interpret these graphs helps you see the trends in data over time, making it easier to predict or explain real-world changes.

Algebraic Solution

An algebraic solution to a system of equations involves manipulating the equations using algebraic methods to find precise values for the unknowns. In our exercise, the lengthier algebraic process involves equating the expressions for \(y\), and solving for \(x\):

• Set the equations equal: \(-0.04x + 5.1 = -0.31x + 5.3\).
• To make solving easier, bring terms with \(x\) to one side and constant terms to another, resulting in: \(0.27x = 0.2\).
• Finally, divide each side to isolate \(x\), giving you \(x = \frac{0.2}{0.27} \approx 0.74\), which rounds to \(x = 1\).
• Use this \(x\) value in the original equation to find \(y\).

Once you substitute \(x = 1\) back into either linear equation, you encounter \(y = 5.06\), which rounds to \(y = 5\). This whole process confirms the year and percent when both heating methods were equally used, thereby resolving the given system algebraically.