Problem 56 - A Graphical Approach to Precalculus with Limits

Question

As of \(2017,\) the top three beauty brands on Facebook (L'Oréal, Dove, and NIVEA) had a combined 74 million followers. L'Oréal and Dove had the same number of followers, and NIVEA had 7 million fewer followers. How many followers did each brand have?

Step-by-Step Solution

Answer

L'Oréal and Dove each have 27 million followers; NIVEA has 20 million.

Step 1: Define Variables

Let \( x \) represent the number of followers that L'Oréal and Dove each have, since they had the same number of followers. Let \( x - 7 \) represent the number of followers that NIVEA has, as it had 7 million fewer followers.

Step 2: Set Up the Equation

Based on the problem, the equation for the combined followers can be written as: \[ x + x + (x - 7) = 74 \]

Step 3: Combine Like Terms

Combine the \( x \) terms to simplify the equation: \[ 3x - 7 = 74 \]

Step 4: Solve for x

Add 7 to both sides to isolate the \( 3x \) term: \[ 3x = 81 \]. Then divide both sides by 3: \[ x = 27 \]. This means L'Oréal and Dove each have 27 million followers.

Step 5: Calculate NIVEA's Followers

Substitute \( x = 27 \) into the expression for NIVEA's followers: \[ x - 7 \]. Thus, NIVEA has \( 27 - 7 = 20 \) million followers.

Key Concepts

Variables in Mathematics

Variables are a fundamental concept in mathematics and essential when solving algebraic equations. They allow us to represent unknown or varying quantities with symbols, such as the letter \( x \). This makes it easier to formulate and solve problems. In the original problem, variables are used to represent the number of followers that L'Oréal and Dove have. We used \( x \) for these brands since we know they had an equal number of followers. For NIVEA, which had fewer followers, we use \( x - 7 \) to capture this difference. Using variables helps in creating relationships between different quantities and simplifies complex descriptions into manageable equations. Some useful tips when working with variables include:

• Choose a letter that makes sense for what you are representing to avoid confusion later in your solution.
• Write down what each variable represents, just like we did with \( x \) and \( x - 7 \).
• Be consistent with the variables throughout your calculations to maintain clarity.

Equation Simplification

Equation simplification is the process of making an equation easier to solve by combining like terms and eliminating unnecessary components. This process often involves rewriting equations in their simplest form without altering the equality. In our exercise, we start by translating the word problem into the equation: \( x + x + (x - 7) = 74 \). Then we simplify by combining the \( x \) terms, resulting in \( 3x - 7 = 74 \). This simplification step is crucial as it directly influences how easily we can solve for \( x \). Here are some strategies for simplifying equations:

• Combine like terms: Add or subtract terms with the same variable.
• Remove constant terms: Move constants to the other side of the equation if needed.
• Re-check simplification: Ensure no terms are left unsimplified for accurate results.

Simplification helps in visualizing the problem better and understanding the relationships involved.

Arithmetic Operations

Arithmetic operations, including addition, subtraction, multiplication, and division, are the building blocks of solving equations. In algebra, these operations help us manipulate equations to find the value of unknown variables.In our solution, we processed the simplified equation \( 3x - 7 = 74 \) by applying these operations step by step.

• First, we added 7 to both sides to isolate the term \( 3x \): \( 3x = 81 \).
• Then, we divided both sides by 3 to solve for \( x \): \( x = 27 \).

Each arithmetic operation serves a distinct role in isolating variables to find their true value. When performing these operations, it's important to:

• Keep the equation balanced: Operations on one side must also be applied to the other.
• Follow the order of operations: Perform multiplication and division before addition and subtraction when needed.
• Check your result: Substitute back to ensure the solution satisfies the original equation.

These operations are crucial not only for this exercise but also for a wide range of algebraic problem-solving scenarios.