Problem 99 - A Graphical Approach to Precalculus with Limits

Question

Advertising Spending In \(2015,\) AT\&T and Verizon spent a combined \(\$ 1212\) million on network TV advertising. AT\&T spent \(\$ 250\) million more than Verizon. (a) Write a system of equations whose solution gives the spending of each company, in millions of dollars. Let \(x\) be the amount spent by AT\&T and \(y\) be the amount spent by Verizon. (b) Solve the system of equations. (c) Interpret the solution.

Step-by-Step Solution

Answer

AT&T spent $731 million, and Verizon spent $481 million.

Step 1: Define the Variables

Let \( x \) represent the amount spent by AT\&T and \( y \) represent the amount spent by Verizon.

Step 2: Establish the Equations

According to the problem, AT\&T and Verizon together spent \( 1212 \) million dollars on network TV advertising, and AT\&T spent \( 250 \) million dollars more than Verizon. From this information, we can set up the following system of equations: \( x + y = 1212 \) and \( x = y + 250 \).

Step 3: Substitute and Solve for y

To solve for \( y \), we can substitute the second equation into the first. So, substitute \( x = y + 250 \) into \( x + y = 1212 \): \( (y + 250) + y = 1212 \). Simplify and solve for \( y \): \[ 2y + 250 = 1212 \] Subtract \( 250 \) from both sides: \[ 2y = 962 \] Divide by \( 2 \): \( y = 481 \).

Step 4: Solve for x

Using \( y = 481 \) from the previous step, substitute back into the second equation to find \( x \): \( x = y + 250 = 481 + 250 \). Thus, \( x = 731 \).

Step 5: Interpret the Solution

The solution \( x = 731 \) and \( y = 481 \) means that AT\&T spent \( 731 \) million dollars and Verizon spent \( 481 \) million dollars on network TV advertising in 2015.

Key Concepts

Linear Equations

Linear equations are mathematical statements that show the equality between two expressions, where each term is either a constant or the product of a constant and a single variable. They take the form \( ax + b = c \) or similar variations, where \( x \) and \( y \) are variables, and \( a \), \( b \), and \( c \) are constants. In the context of this exercise, we're dealing with two linear equations that represent the situation of advertising spending by AT&T and Verizon.

• The first equation, \( x + y = 1212 \), accounts for the total spending between both companies.
• The second equation, \( x = y + 250 \), reflects that AT&T spent $250 million more than Verizon.

These equations help us understand the specific conditions of this scenario, setting up a framework that can be solved using methods for systems of equations.

Substitution Method

The substitution method is a common way to solve systems of equations where one equation is rewritten to express one variable in terms of the other. This expression is then used to replace the variable in the second equation, simplifying the system into a single equation in one unknown variable.
Here's how it works in this situation:

• Start with the equations: \( x + y = 1212 \) and \( x = y + 250 \).
• Substitute \( x = y + 250 \) into the first equation to eliminate \( x \): \((y + 250) + y = 1212\).
• Simplify to find \( y \): \(2y + 250 = 1212\) becomes \(2y = 962\).
• Divide by 2 to get \( y = 481 \).

Use the value of \( y \) to find \( x \): substitute \( y = 481 \) back into \( x = y + 250 \), giving \( x = 731 \). This method helps make solving systems of equations straightforward by gradually reducing them to one variable.

Interpretation of Solutions

Interpreting the solution of a system of equations is about understanding what the variables' values represent in the context of the problem. After solving the equations, we have found that \( x = 731 \) and \( y = 481 \).

• Here, \( x = 731 \) indicates that AT&T spent \(731 million on network TV advertising.
• \( y = 481 \) tells us that Verizon spent \)481 million.

The solution reflects the initial conditions set by the problem, confirming that the total spending was \(1212 million and that AT&T spent exactly \)250 million more than Verizon. Interpreting these numbers helps to reinforce the understanding that solutions to equations are more than just number-crunching—they offer insights into real-world scenarios. Understanding this helps bridge the gap between mathematical theory and practical application.