Problem 85 - A Graphical Approach to College Algebra

Question

From January to June \(2012,\) Samsung and Apple spent a combined 293 million dollars on media. Apple spent 93 million dollars more than Samsung. (a) Write a system of equations whose solution gives the spending of each media company, in millions of dollars. Let \(x\) be the amount spent by Apple and \(y\) be the amount spent by Samsung. (b) Solve the system of equations. (c) Interpret the solution.

Step-by-Step Solution

Answer

Apple spent 193 million dollars; Samsung spent 100 million dollars.

Step 1: Define the Variables

Let \( x \) represent the amount spent by Apple in millions of dollars, and \( y \) represent the amount spent by Samsung in millions of dollars.

Step 2: Set Up the System of Equations

Based on the problem statement, we have two conditions: 1. Samsung and Apple spent a combined total of 293 million dollars. This gives us the equation: \( x + y = 293 \).2. Apple spent 93 million dollars more than Samsung, which gives us a second equation: \( x = y + 93 \).

Step 3: Substitute the Second Equation into the First

Substitute \( x = y + 93 \) from the second equation into the first equation \( x + y = 293 \). This results in: \( (y + 93) + y = 293 \).

Step 4: Simplify and Solve for y

Simplify the equation from Step 3: \( y + 93 + y = 293 \) becomes \( 2y + 93 = 293 \).Subtract 93 from both sides: \( 2y = 200 \).Divide both sides by 2: \( y = 100 \). Thus, Samsung spent 100 million dollars.

Step 5: Solve for x

Use the value of \( y \) to find \( x \). Substitute \( y = 100 \) into the equation \( x = y + 93 \): \( x = 100 + 93 \) results in \( x = 193 \). Thus, Apple spent 193 million dollars.

Step 6: Interpret the Solution

Apple spent 193 million dollars on media, while Samsung spent 100 million dollars from January to June 2012. Together, they spent 293 million dollars, satisfying the conditions set in the problem statement.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and involve variables raised to the power of one. In real-world scenarios, they help us model relationships between quantities. In this exercise, we're given the total media spending of Samsung and Apple as well as the difference in their spending.

We can describe such situations using linear equations:

• The total spending equation: \( x + y = 293 \)
• The difference in spending equation: \( x = y + 93 \)

Here, \( x \) represents the amount Apple paid, and \( y \) represents Samsung's spending.

These equations provide a clear, structured way to represent the relationship between the two companies' expenditures. They act as a mathematical framework that we can solve to gain insights into the problem.

Substitution Method

The substitution method is a strategy for solving systems of linear equations. It involves solving one equation for one variable and substituting this value into another equation.

Let's break this down:

• First, solve one of the equations for one variable. In our exercise, we already have \( x = y + 93 \), which expresses \( x \) in terms of \( y \).
• Next, substitute \( (y + 93) \) for \( x \) in the total spending equation: \( x + y = 293 \).

This substitution gives us a single equation in one variable: \( (y + 93) + y = 293 \).

The substitution method simplifies the problem, making it easier to solve by reducing the number of unknowns. It's a handy technique for problems where one equation is already solved for a variable, giving us a direct path to the solution.

Solving Equations

Solving equations involves finding the value of unknown variables that make an equation true. After substituting in the exercise, we're left with the equation \( 2y + 93 = 293 \).

Here's how we solve it step-by-step:

• First, simplify by combining like terms: \( 2y + 93 = 293 \).
• Next, isolate \( y \) by subtracting 93 from both sides: \( 2y = 200 \).
• Finally, divide both sides by 2 to solve for \( y \): \( y = 100 \).

Now, we've determined Samsung's spending. To find \( x \), Apple's spending, substitute \( y = 100 \) back into \( x = y + 93 \), resulting in \( x = 193 \).

The process of solving equations is about working systematically. We simplify expressions, isolate terms, and solve for unknowns, ensuring the conditions of the problem are satisfied. By following these steps, we confirm the accuracy and consistency of our solution.