Problem 121 - College Algebra with Modeling and Visualization

Question

The total number of robberies in 2000 and 2001 was \(831,000 .\) From 2000 to 2001 the number of robberies declined by \(15,000 .\) (a) Write a system of equations whose solution represents the number of robberies committed in each of these years. (b) Solve the system symbolically. (c) Solve the system graphically.

Step-by-Step Solution

Answer

2000: 423,000 robberies; 2001: 408,000 robberies.

Step 1: Understand the Problem

We're given the total number of robberies over two years and how many fewer robberies occurred in 2001 compared to 2000. We need first to find out the number of robberies for each individual year.

Step 2: Formulate Equations

Let \( x \) represent the number of robberies in 2000, and \( y \) represent the number of robberies in 2001. We know two things: 1. \( x + y = 831,000 \) (total number) 2. \( x - y = 15,000 \) (difference from 2000 to 2001). These two equations form our system of equations.

Step 3: Solve Symbolically: Add Equations

First, add the two equations to eliminate \( y \) and solve for \( x \):\((x + y) + (x - y) = 831,000 + 15,000 \)\(2x = 846,000 \)\(x = 423,000 \) robberies in 2000.

Step 4: Solve Symbolically: Substitute to Find y

Substitute \( x = 423,000 \) into the first equation to find \( y \):\(423,000 + y = 831,000 \)\(y = 408,000 \) robberies in 2001.

Step 5: Solve Graphically

Graph the equations \( y = 831,000 - x \) and \( y = x - 15,000 \). The solution to the system corresponds to the point where these two lines intersect on a graph. This intersection is at the point (423,000, 408,000).

Key Concepts

Algebraic Solution

When solving a system of linear equations through algebraic methods, we often rely on techniques like substitution or elimination. In this context, we used the method of elimination to start with.
In simple terms, elimination involves either adding or subtracting equations to "eliminate" one of the variables. Here, by adding the two given equations:

• First Equation: \( x + y = 831,000 \)
• Second Equation: \( x - y = 15,000 \)

When we add them, the \( y \) variables cancel each other out, simplifying the system to:

• \( 2x = 846,000 \)
• Next, divide by 2 to find \( x \): \( x = 423,000 \).
• Substitute this back into the first equation to find \( y \): \( 423,000 + y = 831,000 \), giving \( y = 408,000 \).

This algebraic solution reveals the number of robberies for each year clearly. By approaching it step-by-step, it minimizes potential errors while making the problem manageable.

Graphical Solution

Graphical solutions provide a visual approach to solving systems of linear equations. By converting equations into line equations, we can graph them to find their point of intersection.
For our example, we transform the given equations:

• First Equation rearranged: \( y = 831,000 - x \)
• Second Equation rearranged: \( y = x - 15,000 \)

When graphing these two lines:

• The slope of the first line is negative, indicating it goes downwards.
• The slope of the second line is positive, going upwards.

The point where these two lines intersect represents the solution to our system of equations. In this example, the lines meet at \( (423,000, 408,000) \). This graphical solution confirms our algebraic result, showing that both methods can effectively solve the problem but in different ways.

Problem-Solving in Algebra

Solving algebraic equations and problems requires not just understanding equations but also translating word problems into mathematical terms. Problem-solving in algebra involves several critical steps:

• Understanding the problem: Identify what is known and what needs to be determined.
• Formulating equations: Convert the word problems into algebraic equations using variables.
• Choosing a solution method: Deciding whether to use substitution, elimination, or graphical representation.
• Evaluating: Check if the solution satisfies all original conditions.

The problem of finding the number of robberies in two different years not only illustrated the application of algebraic principles but further stressed the importance of selecting the appropriate method. Emphasize practicing different techniques and reviewing their applications, as this develops a more robust problem-solving skillset in algebra.