Question

U.S. Advertising Share A report showed that Internet ads accounted for \(19 \%\) of all U.S. advertisement spending when print ads (magazines and newspapers) accounted for \(26 \%\) of the spending. The report further showed that Internet ads accounted for \(35 \%\) of all advertisement spending when print ads accounted for \(16 \%\) of the spending. (a) Write a linear equation that relates that percent \(y\) of print ad spending to the percent \(x\) of Internet ad spending. (b) Find the intercepts of the graph of your equation. (c) Do the intercepts have any meaningful interpretation? (d) Predict the percent of print ad spending if Internet ads account for \(39 \%\) of all advertisement spending in the United States.

Short answer

y = -\frac{5}{8}x + \frac{303}{8}. Intercepts: y-intercept \( \frac{303}{8} \), x-intercept 60.6. Predicted print ad spending: 13.5%.

Step-by-step solution

- Set up variables

Define the variables for the problem. Let the percent of Internet ad spending be represented by x and the percent of print ad spending be represented by y.

- Create point pairs

Create point pairs using the given percentages. From the information, we have two points: (19, 26) and (35, 16).

- Determine the slope

Use the slope formula to find the slope between the two points. The slope formula is \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). Plug in the given points: \(m = \frac{{16 - 26}}{{35 - 19}} = \frac{{-10}}{{16}} = -\frac{5}{8}\).

- Write the equation in point-slope form

Use one of the points and the slope to write the equation in point-slope form: \(y - y_1 = m(x - x_1)\). Using the point (19, 26) and the slope \(-\frac{5}{8}\), we get: \( y - 26 = -\frac{5}{8}(x - 19)\).

- Convert to slope-intercept form

Simplify the equation from the previous step to get it into slope-intercept form (\(y = mx + b\)). Begin by distributing: \( y - 26 = -\frac{5}{8}x + \frac{95}{8}\). Then isolate y: \( y = -\frac{5}{8}x + \frac{95}{8} + 26\) To combine the constants, find a common denominator (8): \( y = -\frac{5}{8}x + \frac{95}{8} + \frac{208}{8} = -\frac{5}{8}x + \frac{303}{8}\). Thus, the equation is \( y = -\frac{5}{8}x + \frac{303}{8}\).

- Find the intercepts

To find the y-intercept, evaluate the equation when x=0: \( y = -\frac{5}{8}(0) + \frac{303}{8} = \frac{303}{8}\). For the x-intercept, set y=0 and solve for x: \( 0 = -\frac{5}{8}x + \frac{303}{8}\). Multiply through by 8: \( 0 = -5x + 303\), then solve: \( 5x = 303\), so \( x = 60.6\). The y-intercept is \( \frac{303}{8}\) and the x-intercept is 60.6.

- Interpret the intercepts

The y-intercept represents the predicted print ad spending when Internet ad spending is 0%. The x-intercept represents the predicted Internet ad spending when print ad spending is 0%.

- Predict for 39%

Substitute x=39 into the equation to predict the print ad spending: \( y = -\frac{5}{8}(39) + \frac{303}{8}\) Simplify: \( y = -\frac{195}{8} + \frac{303}{8} = \frac{108}{8} = 13.5\). So, if Internet ads account for 39%, print ad spending is predicted to be 13.5%.

Key concepts

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations. It is written as:
\(y = mx + b\).
Here, \(m\) is the slope of the line, and \(b\) is the y-intercept.
In our exercise, we transformed the equation from point-slope form to slope-intercept form.
This helps us easily identify the slope and the y-intercept.

Why is this useful?
The slope-intercept form makes it easy to graph the line and understand its behavior.
The slope \(m\) tells us how steep the line is.
The y-intercept \(b\) shows where the line crosses the y-axis.

For example, in our problem, the slope-intercept form of the equation is:
\(y = -\frac{5}{8}x + \frac{303}{8}\).
Here, the slope \(-\frac{5}{8}\) shows the decrease in print ad spending per unit increase in Internet ad spending.
The y-intercept \( \frac{303}{8}\) is the expected print ad spending when Internet ad spending is 0%.
Understanding this form helps answer parts (b) and (c) of the exercise effectively.

Point-Slope Form

Point-slope form is another way to write the equation of a line. It is written as:
\( y - y_1 = m(x - x_1) \).
Here, \( (x_1, y_1) \) is a point on the line, and \(m\) is the slope.

How does it help?
Point-slope form is particularly useful when you have a point and the slope, but not the intercepts.
This form directly uses the known point to describe the line.

In our exercise, we used the point-slope form to start our equation:
\( y - 26 = -\frac{5}{8}(x - 19) \).
Here, (19, 26) is a point from our data, and \(-\frac{5}{8}\) is the slope we calculated.

Why is this step important?
Starting with point-slope form is helpful for transitioning to slope-intercept form.
It allows us to use real data points to create our linear model.
This step is foundational in building an accurate representation of the relationship in the data.

Intercepts

Intercepts are points where the line crosses the axes. There are two types:

• Y-intercept: where the line crosses the y-axis. It is found by setting \(x = 0\).
• X-intercept: where the line crosses the x-axis. It is found by setting \(y = 0\).

How do you find them?
To find the y-intercept, substitute \(x = 0\) into the equation:
\( y = -\frac{5}{8}(0) + \frac{303}{8} = \frac{303}{8} \).
So, the y-intercept is \( \frac{303}{8} \).
To find the x-intercept, set \(y = 0\) and solve for \(x\):
\( 0 = -\frac{5}{8}x + \frac{303}{8} \).
Multiplying through by 8 gives \( 0 = -5x + 303\). Solving, we get \( x = 60.6 \).
So, the x-intercept is 60.6.

What do they mean?
In our context, intercepts help us understand the predictions.
The y-intercept shows predicted print ad spending when Internet ad spending is zero.
The x-intercept shows predicted Internet ad spending when print ad spending is zero.
These insights help in interpreting the relationships between these two types of ad spending.