Problem 45 - Elementary Algebra Within Reach

Question

Describe and correct the error in writing the system of linear equations for the problem below. Do not solve the system. The sum of two numbers is 42 , and the larger number is 3 less than twice the smaller number. $$ \begin{aligned} x+y &=42 \\ 2 x-y &=-3 \end{aligned} $$

Step-by-Step Solution

Answer

The error is in the second equation, which should be \(y = 2x - 3\) to correctly represent 'the larger number is 3 less than twice the smaller number'. The correct system of equations is \(x + y = 42\) and \(y = 2x - 3\).

Step 1: Identify the Error

The issue lies in the second equation. The equation \(2x - y = -3\) doesn't properly translate the statement 'the larger number is 3 less than twice the smaller number'.

Step 2: Analyze the Problem Statement

Assuming x is the smaller number and y is the larger number, the correct translation would be 'y (the larger number) is equal to twice x (smaller number) minus 3', or formally written as \(y = 2x - 3\).

Step 3: Correct the System of Equations

The correct system of equations reflecting the problem statement would therefore be: \(x + y = 42\) and \(y = 2x - 3\).

Key Concepts

Common Algebraic Mistakes

When solving systems of linear equations, making algebraic mistakes is quite common, especially when dealing with complex problem statements. Here are some typical errors and how to avoid them:

• Incorrectly translating word problems into mathematical equations. This often happens when the language nuances in a problem statement aren't fully understood, leading to misinterpretation.
• Mixing up the roles of variables, such as mistakenly designating what each variable represents, which can lead to incorrect equations.
• Math symbol errors, like using "=" instead of "-" or failing to reverse the inequality sign when dividing by a negative.

To correct these mistakes, always carefully read the problem, double-check each step, and verify that your final equations correctly reflect the statements given in the problem.

Translating Word Problems into Equations

Translating word problems into equations is a crucial skill in solving algebra systems. The main challenge is converting the linguistic clues into mathematical symbols. Here are some tips to do it correctly:

• Identify and define what each variable represents. Assign meaningful variables such as 'size', 'time', or 'cost' to correspond to what you're solving for.
• Break down the problem statement into smaller parts. For example, separate the information about sums, differences, products, or ratios, and construct each part into individual equations.
• Look for keywords. Words like "sum," "difference," "twice," and "less" provide hints for forming equations.
• Check your equations by substituting back into the word problem. Ensure that the formulations align with the problem's context and the relationships described.

By practicing these strategies, translating word problems becomes less daunting and more straightforward.

Writing Linear Equations Correctly

Writing linear equations correctly is essential for accurately representing mathematical relationships. Here’s how you can achieve precision in your equations:

• Ensure that each variable is correctly positioned according to the problem's specifications. For instance, if 'y' is supposed to be a function of 'x', ensure it appears correctly on one side of the equation.
• Maintain equation balance by performing equivalent operations on both sides. Whether adding, subtracting, or multiplying, the balance of the equation must remain.
• Replace phrases from the problem with appropriate mathematical symbols. For example, "3 less than twice the number" translates to "-3" after "2x" in an equation, illustrating the importance of sequencing.
• Double-check for mathematical consistency. Confirm that both equations in a system accurately match the conditions set by the problem.

Consistently applying these practices enhances your skills in writing equations and helps to ensure they reflect the scenarios you intend to describe.