Problem 65 - Intermediate Algebra

Question

Area Codes. The entire state of Alaska (except for the small community of Hyder) has just one telephone area code. The same is true for the state of Hawaii. If we let \(A=\) the area code of Alaska and \(H=\) the area code of Hawaii, we can find \(A\) and \(H\) by solving the system: $$ \left\\{\begin{array}{l} A+H=1,715 \\ A-H=99 \end{array}\right. $$ What is the area code of each state?

Step-by-Step Solution

Answer

Alaska's area code is 907, and Hawaii's area code is 808.

Step 1: Understanding the System of Equations

The given system of equations is: \( A + H = 1,715 \) and \( A - H = 99 \). These equations define the relationships between the area codes of Alaska \( (A) \) and Hawaii \( (H) \). Our goal is to solve these equations simultaneously to find the values of \( A \) and \( H \).

Step 2: Solving for A

Add the two equations together to eliminate \( H \):\[(A + H) + (A - H) = 1,715 + 99\]This simplifies to \( 2A = 1,814 \). Divide both sides by 2 to isolate \( A \):\[A = \frac{1,814}{2} = 907\]

Step 3: Solving for H

Now that we know \( A = 907 \), substitute \( A \) into the first equation, \( A + H = 1,715 \):\[907 + H = 1,715\]Subtract 907 from both sides to find \( H \):\[H = 1,715 - 907 = 808\]

Step 4: Verify the Solution

Substitute \( A = 907 \) and \( H = 808 \) back into the second equation, \( A - H = 99 \):\[907 - 808 = 99\]The calculations are correct, confirming that the solutions are consistent with both equations.

Key Concepts

Simultaneous Equations

A system of simultaneous equations involves two or more equations that share common variables. In our problem, we are dealing with two equations that involve the variables \( A \) and \( H \), representing the area codes for Alaska and Hawaii respectively. Solving simultaneous equations means finding the values of the variables that satisfy all given equations at the same time.

Here are a few points to remember about simultaneous equations:

• They can be linear or nonlinear, but in many practical applications, such as the one described here, they are linear.
• The solution to a system of two linear equations usually results from finding the intersection point of the two lines these equations represent on a graph. Here, we use algebraic methods instead.
• The goal is to find a common set of variable values that makes all equations true simultaneously.

In this exercise, the two equations \( A + H = 1,715 \) and \( A - H = 99 \) are both linear. They model a simple relationship between two variables, and their solution provides the area codes for the states involved.

Algebraic Solution

The algebraic solution for simultaneous equations involves manipulating the equations to isolate and solve for one or more variables. This is done using algebraic operations such as addition, subtraction, multiplication, and division. In our given problem, we are tasked with solving for \( A \) and \( H \).

Initially, notice that each equation consists of two terms involving \( A \) and \( H \). By carefully adding or subtracting equations, you can eliminate one of the variables to easily find the value of the other. For instance, adding \( A + H = 1,715 \) and \( A - H = 99 \) eliminates \( H \), leaving you with \( 2A = 1,814 \).

Once \( A \) is found, substitute its value back into one of the original equations to solve for \( H \).

Why use this method?

• It's logical and direct, especially when equations are set up for easy elimination of one variable.
• It avoids graphing, which might be less practical for larger or more complex systems of equations.

This kind of algebraic solving is a fundamental skill useful across many areas of mathematics and real-world problem-solving.

Verification of Solution

Verification of a solution in a system of equations is an essential step to ensure the accuracy and consistency of the findings. Once potential solutions for \( A \) and \( H \) are calculated, it is crucial to substitute these back into the original equations to verify they satisfy both.

For our system, we found \( A = 907 \) and \( H = 808 \). Substituting these values into both equations should maintain the stated equality. In the case of the first equation, \( A + H = 1,715 \), substituting gives \( 907 + 808 = 1,715 \), which holds true. For the equation \( A - H = 99 \), substituting gives \( 907 - 808 = 99 \), also confirming the solution.

Key reasons for verification:

• It confirms that the solution is correct and complies with all given equations.
• It helps detect any algebraic errors made during calculation.

Verification ensures that not only are calculations correct, but interpretations of the mathematical model also align with the problem's requirements, leading to robust and reliable conclusions.