Problem 55 - A Graphical Approach to Precalculus with Limits

Question

As of 2017 , the total combined number of monthly users of Facebook, Instagram, and Twitter was 223 million. The combined number of users of Instagram and Twitter was 23 million less than the number of users of Facebook. There were 18 million more users of Instagram than users of Twitter. How many users of each social network were there?

Step-by-Step Solution

Answer

Facebook: 123M, Instagram: 59M, Twitter: 41M.

Step 1: Define Variables

Let's define the variables: let \( F \) be the number of Facebook users, \( I \) be the number of Instagram users, and \( T \) be the number of Twitter users.

Step 2: Set Up Equations

From the problem, we have the following equations: 1) \( F + I + T = 223 \)2) \( I + T = F - 23 \)3) \( I = T + 18 \) These equations come from the relations given in the problem statement.

Step 3: Substitute Into Equations

Substitute \( I = T + 18 \) from equation 3 into equation 2: \( T + 18 + T = F - 23 \), simplifies to \( 2T + 18 = F - 23 \), further simplifies to \( F = 2T + 41 \).

Step 4: Substitute F and I into First Equation

Now, substitute \( F = 2T + 41 \) and \( I = T + 18 \) into the first equation: \((2T + 41) + (T + 18) + T = 223 \).

Step 5: Solve for T (Twitter Users)

This simplifies to \( 4T + 59 = 223 \). Subtract 59 from both sides: \( 4T = 164 \)Divide by 4: \( T = 41 \).So, there are 41 million Twitter users.

Step 6: Calculate I (Instagram Users)

Use \( I = T + 18 \) to find \( I \): \( I = 41 + 18 = 59 \).So, there are 59 million Instagram users.

Step 7: Calculate F (Facebook Users)

Use \( F = 2T + 41 \) to find \( F \): \( F = 2(41) + 41 = 82 + 41 = 123 \).So, there are 123 million Facebook users.

Key Concepts

Variables and Expressions

In algebra, variables and expressions are fundamental components used to solve problems. Variables are symbols, often letters, used to represent unknown values. Expressions are combinations of these symbols and numbers using mathematical operations like addition or subtraction.
In our problem, we use variables to represent the number of users on different social networks:

• Facebook users are represented by the variable \( F \).
• Instagram users are represented by \( I \).
• Twitter users are represented by \( T \).

Using variables allows us to form mathematical expressions based on the relationships described in the problem. For example, the statement "the combined number of users of Instagram and Twitter was 23 million less than the number of users of Facebook" is transformed into the expression \( I + T = F - 23 \).
Expressions like this are the backbone of setting up equations that can be solved to find these unknown values.

Solving Equations

To find the solution to our problem, we need to use the equations we've derived from the problem's narrative. Solving equations involves finding the values of variables that make the equations true. The process often includes:

• Substituting values or other expressions
• Combining like terms
• Isolating a variable

In our case, we first set up the following key equations:

• \( F + I + T = 223 \)
• \( I + T = F - 23 \)
• \( I = T + 18 \)

Next, we substitute and manipulate these equations to isolate the variables. By substituting \( I = T + 18 \) into the second equation, and then solving for \( F \), we continue to simplify and strategically substitute until each variable is determined. This step-by-step process exemplifies systematic solving of equations.

Substitution Method

The substitution method is a popular and effective strategy for solving systems of equations. It involves solving one of the equations for a variable and then substituting that expression into the other equations.
In this problem, we used substitution when solving for \( I \) from \( I = T + 18 \) and plugged it into another equation. This step leads to a new equation solely in terms of \( T \) and \( F \):

• From \( I = T + 18 \) and \( I + T = F - 23 \), we substitute to get: \( 2T + 18 = F - 23 \).

Rearranging that gives us \( F = 2T + 41 \). The substitution method enables us to break down complex systems into more manageable expressions, eventually leading us to a straightforward arithmetic equation that can be directly solved.
This approach reduces potential errors and helps double-check each step efficiently.