Problem 77 - A Graphical Approach to College Algebra

Question

The total spending on Black Friday during 2011 and 2012 was 1858 million dollars. From 2011 to \(2012,\) spending increased by 226 million dollars. (Source: www.marketingcharts.com) (a) Write a system of equations whose solution represents the Black Friday spending in each of these years. Let \(x\) be the amount spent in 2012 and \(y\) be the amount spent in 2011. (b) Solve the system. (c) Interpret the solution.

Step-by-Step Solution

Answer

2011 spending: $816 million, 2012 spending: $1042 million.

Step 1: Define Variables and Set Up Equations

Let the amount spent in 2011 be \( y \) and the amount spent in 2012 be \( x \). According to the problem, the total spending for both years was 1858 million dollars, leading to the equation: \[ x + y = 1858. \]Further, spending in 2012 increased by 226 million dollars compared to 2011, giving us another equation: \[ x = y + 226. \]

Step 2: Substitution Method to Solve System

Use the second equation to express \( x \) in terms of \( y \):\[ x = y + 226. \]Substitute \( x = y + 226 \) into the first equation:\[ (y + 226) + y = 1858. \]This simplifies to:\[ 2y + 226 = 1858. \]

Step 3: Solve for y

Solve the equation \( 2y + 226 = 1858 \) for \( y \):1. Subtract 226 from both sides: \[ 2y = 1858 - 226. \]2. Simplify the right side: \[ 2y = 1632. \]3. Divide both sides by 2: \[ y = 816. \]

Step 4: Solve for x

Substitute \( y = 816 \) back into the equation \( x = y + 226 \):\[ x = 816 + 226. \]Simplify:\[ x = 1042. \]

Step 5: Verification

Verify the solution by substituting \( x = 1042 \) and \( y = 816 \) back into the original equations:1. Check total spending: \[ 1042 + 816 = 1858. \]2. Check increase from 2011 to 2012: \[ 1042 = 816 + 226. \]Both equations are satisfied, so our solution is correct.

Step 6: Interpretation

The solution \( x = 1042 \) and \( y = 816 \) means that in 2011, the spending was 816 million dollars and in 2012, it was 1042 million dollars. This reflects an increase of 226 million dollars in spending from 2011 to 2012.

Key Concepts

Algebraic Problem Solving

Algebraic problem solving is a critical skill for tackling real-world problems and for using mathematics as a tool to communicate complex ideas. In algebraic problem solving, the first step often involves translating a word problem into a mathematical form. This involves identifying the unknowns and defining variables to represent them.
For instance, in the exercise we are examining, we need to find the amount of money spent on Black Friday in two different years. We set variables where the unknowns were chosen logically: Letting \( x \) be the expenditure for 2012 and \( y \) for 2011 allows us to form our mathematical equations based on the relationships given in the problem. Once the variables are set, you then develop equations by looking for words like 'total', 'difference', or 'more than', which will guide how you combine these variables.
Algebraic equations are the backbone of understanding how different values relate to each other in a mathematical manner. Through them, we can predict, calculate, and understand different scenarios.

Substitution Method

The substitution method is a technique employed to solve systems of equations. It's especially useful when one of the equations is simple enough to solve for one variable in terms of the other. This method consists of three main steps:

• First, solve one of the equations for one of the variables. Preferably the one which appears to simplify easily.
• Next, substitute this expression into the other equation. This changes the two-variable equation into a single-variable equation.
• Finally, solve the resulting equation to find the value of one variable and use it to find the value of the other variable.

By utilizing the substitution method in our exercise, we first derived the expression \( x = y + 226 \) from the second equation to isolate \( x \). Then, we replaced \( x \) in the first equation \( x + y = 1858 \) with \( y + 226 \), which allowed us to solve for \( y \). This streamlined the process into simple arithmetic.

Equation Interpretation

Interpreting equations is the final and crucial step in solving systems of equations, as it validates the logical consistency within the context of the problem. After obtaining the solution for both variables, we need to verify that they make sense in a real-world environment.
In our example, after solving for \( x \) and \( y \), where \( y = 816 \) and \( x = 1042 \), we interpreted these values as the spending amounts for 2011 and 2012, respectively. To ensure accuracy, we substituted back into the original equations: checking that \( x + y = 1858 \) and \( x = y + 226 \). Both conditions held true, confirming that our solution was correct.
This interpretation step goes beyond simple math. It's about understanding and conveying the meaning of the solution, verifying its feasibility, and communicating the results effectively. It ensures that mathematical solutions effectively reflect real-world situations, aligning numbers with their physical significance.