Problem 62 - Algebra A Combined Function

Question

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In 2008 , there were approximately 945 thousand restaurants in the United States. In 2004 , there were 875 thousand restaurants. (Source: National Restaurant Association). a. Assume the relationship between years past 2004 and the number of restaurants is linear over this period. Write an equation describing the relationship between time and the number of restaurants. Use ordered pairs of the form (years past \(2004,\) numbers of restaurants in thousands). b. Use this equation to predict the number of eating establishments in 2012 .

Step-by-Step Solution

Answer

The linear equation is \( y = 17.5x + 875 \). Predicted restaurants in 2012: 1015 thousand.

Step 1: Identify Ordered Pairs

First, identify the ordered pairs based on the information given. In 2004, there were 875,000 restaurants, so the first point is \( (0, 875) \). In 2008, there were 945,000 restaurants, so the second point is \( (4, 945) \).

Step 2: Calculate the Slope

To find the slope \( m \) of the linear equation, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging in the values from our points: \( m = \frac{945 - 875}{4 - 0} = \frac{70}{4} = 17.5 \).

Step 3: Write the Equation in Point-Slope Form

Use the point-slope formula \( y - y_1 = m (x - x_1) \) with a point \((0, 875)\) and slope \( m = 17.5 \). It becomes \( y - 875 = 17.5(x - 0) \).

Step 4: Convert to Slope-Intercept Form

Solve for \( y \) from the point-slope form to get it in slope-intercept form, \( y = mx + b \). Thus, \( y = 17.5x + 875 \). This is the equation describing the relationship.

Step 5: Make a Prediction for 2012

To predict the number of restaurants in 2012, calculate \( x = 2012 - 2004 = 8 \). Substitute \( x = 8 \) into the equation: \( y = 17.5(8) + 875 \).

Step 6: Calculate the Result

Solve for \( y \): \( y = 17.5 \times 8 + 875 = 140 + 875 = 1015 \). Thus, the prediction for the number of restaurants in 2012 is 1015 thousand.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed. They have a constant rate of change and are usually expressed in the form of \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, where the line crosses the y-axis. Linear equations help us understand relationships between two variables, making them crucial in predicting trends and analyzing data.

In our example, we create a linear equation to describe the relationship between the years past 2004 and the number of restaurants in the United States. This equation allows us to visualize and understand how the number of restaurants changes over time, which is important for making future predictions.

Slope Calculation

The slope of a line represents its steepness and direction. A positive slope means the line inclines upwards as it moves from left to right, while a negative slope indicates a downward inclination. The formula to calculate the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In our exercise, the ordered pairs were \((0, 875)\) and \((4, 945)\). Substituting these into the formula gives us:

• \( m = \frac{945 - 875}{4 - 0} = \frac{70}{4} = 17.5 \)

This calculated slope of 17.5 tells us that approximately 17.5 thousand restaurants were added each year from 2004 to 2008, on average.

Point-Slope Form

The point-slope form of a linear equation is a handy way to write the equation if you know a point on the line and the slope. This form is expressed as \(y - y_1 = m(x - x_1)\). In our problem, we used the point \((0, 875)\) and our calculated slope of 17.5. Substituting these values, the equation becomes:

• \(y - 875 = 17.5(x - 0)\)

This form highlights how the y-value changes based on the x-value. When simplified to the slope-intercept form, it gives a complete view of the line's behavior over time.

• \(y = 17.5x + 875\)

This simplification to y-intercept form is useful for easily predicting future values, as it clearly shows both the slope and starting point.

Predicting Values

Predicting values using a linear equation involves substituting the given x-value into the equation and solving for y. This is beneficial for forecasting future trends based on current data. In our exercise, we predicted the possible number of restaurants in 2012, given our linear equation.

• We calculated \(x = 2012 - 2004 = 8\).
• Then, we substituted \(x = 8\) into the equation: \(y = 17.5 \times 8 + 875\).
• This resulted in \(y = 140 + 875 = 1015\).

Thus, based on our linear model, we predicted that there could be 1,015,000 restaurants in 2012. This technique helps in strategic planning and resource allocation by providing a glimpse into possible future scenarios.