Problem 8 - Elementary and Intermediate Algebra

Question

Advertising. In 2008 , U.S. companies spent a total of S1 18.2 billion on newspaper, television, and magazine ads. The total amount spent on television ads was \(\$ 10.8\) billion more than the amount spent on newspaper and magazine ads together. The amount spent on magazine ads was \(\$ 3.5\) billion more than the amount spent on newspaper ads. How much was spent on each form of advertising?

Step-by-Step Solution

Answer

Newspaper: \$34.6333 billion, Magazine: \$38.1333 billion, Television: \$83.5666 billion.

Step 1 Title - Define variables

Let the amount spent on newspaper ads be denoted as \( N \). Define the amount spent on magazine ads as \( M \) and the amount spent on television ads as \( T \).

Step 2 - Create equations based on given information

From the problem, we have three main pieces of information that can be translated into equations:1. \[ T + M + N = 118.2 \]2. \[ T = 10.8 + M + N \]3. \[ M = 3.5 + N \]

Step 3 - Substitute and simplify

Substitute the equation \( M = 3.5 + N \) into the other equations:Replace \( M \) in first equation: \[ T + (3.5 + N) + N = 118.2 \]Simplify the equation: \[ T + 3.5 + 2N = 118.2 \]Next, substitute \( M \) and the value of \( T \) from the second equation: \[ (10.8 + M + N) + (3.5 + N) + N = 118.2 \]

Step 4 - Solve for one variable

Solving the simplified equation from the substitution:\[ 10.8 + 3.5 + 3N = 118.2 \]Combine like terms: \[ 14.3 + 3N = 118.2 \]Isolate \( N \): \[ 3N = 118.2 - 14.3 \]\[ 3N = 103.9 \]\[ N = \frac{103.9}{3} \]\[ N = 34.6333 \]

Step 5 - Solve for remaining variables

Using \( N = 34.6333 \):For \( M \):\[ M = 3.5 + 34.6333 \]\[ M = 38.1333 \]For \( T \):\[ T = 10.8 + 38.1333 + 34.6333 \]\[ T = 83.5666 \]

Key Concepts

linear equations

Linear equations are a type of equation where each term is a constant or the product of a constant and a single variable. These equations represent straight lines when graphed. In the advertising expenditure problem, we deal with three linear equations that describe the relationships between newspaper, magazine, and television ad spending:

• The total ad expenditure on newspaper, television, and magazine ads is given as The algebraic form of this equation is + N + N
  The term N + N is equivalent to This simplified equation is also known as a linear equation. Such equations typically have one or two terms, making them relatively simple to solve. To find the total ad expenditure on each type of advertisement, we need to solve the system of equations we create from these relationships.

variable substitution

Variable substitution is a technique used to simplify equations by replacing one variable with another. This method allows us to reduce the number of variables and make solving the system of equations easier. In our problem, we can substitute the expression for magazine ads, M = 3.5 + N, into the other equations. Here's how we do it:

• Replace M in the first equation: T + (3.5 + N) + N = 118.2
• Simplify the equation: T + 3.5 + 2N = 118.2

This process eliminates the M variable and we now have an equation with only T and N. Doing this makes it easier to manipulate and solve the system of equations step by step.

algebraic manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate and solve for variables. It's a vital skill in solving linear equations and systems of equations. In the advertising expenditure problem, we use algebraic manipulation to combine like terms and isolate a variable. For example, once we've substituted for M, we simplify and solve for N:

• Combine like terms: 10.8 + 3.5 + 3N = 118.2
• Simplify the equation: 14.3 + 3N = 118.2

By isolating N, we subtract 14.3 from both sides:
3N = 103.9
N = \( \frac{103.9}{3} = 34.6333 \)
With this, we've found the amount spent on newspaper ads. We can then use this value of N to find the amounts spent on magazine and television ads through further substitution and manipulation.

system of equations

A system of equations is a set of equations with multiple variables that you solve together. The goal is to find the values of all the variables. In our advertising problem, we have a system of three equations:

• T + M + N = 118.2
• T = 10.8 + M + N
• M = 3.5 + N

We solve this system by using substitution and elimination techniques. We start by substituting M in the equations and then isolate T and N:

• Substitute M: T + (3.5 + N) + N = 118.2
• Simplify to reduce the number of variables: T + 3.5 + 2N = 118.2
• Substitute T from the other equation: (10.8 + M + N) + (3.5 + N) + N = 118.2

Combining these simplified equations allows us to solve the system step by step. By doing so, we can determine the amount spent on each type of advertisement. The key is to systematically substitute and eliminate until each variable is isolated and solved.