Problem 25 - Fundamentals of Mathematics

Question

Four numbers add to \(-2 .\) The second number is three more than twice the negative of the first number. The third number is six less than the first number. The fourth number is eleven less than twice the first number. Find the numbers.

Step-by-Step Solution

Answer

The four numbers are 6, -9, 0, and 1.

Step 1: Define Variables

Let's denote the four numbers as variables. Let the first number be \( x \). Then, the second number can be represented as \( 3 - 2x \), the third number as \( x - 6 \), and the fourth number as \( 2x - 11 \).

Step 2: Create Equation from Given Condition

According to the problem, the sum of the four numbers is \(-2\). Therefore, we can set up the equation as follows:\[ x + (3 - 2x) + (x - 6) + (2x - 11) = -2 \]

Step 3: Simplify and Solve the Equation

Combine like terms in the equation:- Terms with \( x \): \( x - 2x + x + 2x = 2x \) - Constant terms: \( 3 - 6 - 11 = -14 \) The equation now is:\[ 2x - 14 = -2 \]

Step 4: Solve for x

Add 14 to both sides of the equation to isolate the terms with \( x \):\[ 2x = -2 + 14 \]\[ 2x = 12 \]Divide by 2:\[ x = 6 \]

Step 5: Calculate Each Number

Now that \( x = 6 \), substitute \( x \) back into the expressions for each number:- First number: \( x = 6 \) - Second number: \( 3 - 2x = 3 - 2(6) = 3 - 12 = -9 \) - Third number: \( x - 6 = 6 - 6 = 0 \) - Fourth number: \( 2x - 11 = 2(6) - 11 = 12 - 11 = 1 \)

Step 6: Verify the Solution

Check if the sum of the numbers is \(-2\):\[ 6 + (-9) + 0 + 1 = -2 \]Since the sum matches, the solution is correct.

Key Concepts

Variable Representation

In algebraic equations, variable representation is crucial to solve problems efficiently. When trying to find unknown values, we use variables as placeholders. In this exercise, we were asked to find four numbers based on a set of conditions.

We started by representing these unknown numbers with variables. Specifically, we assigned the first number as \( x \). This simple assignment then helped us express the other numbers in terms of \( x \) based on the relationships provided in the problem statement:

• The second number was described as three more than twice the negative of the first number, which translates to \( 3 - 2x \).
• The third number was six less than the first, resulting in \( x - 6 \).
• The fourth number was eleven less than twice the first, giving \( 2x - 11 \).

Knowing how to express conditions through variable representation can make complicated problems more manageable.

Linear Equations

A linear equation is an algebraic expression where each term is either a constant or a product of a constant and a variable. These equations form a straight line when graphed and are one of the simplest types of equations to solve.

In this problem, once we set up our expressions for each of the numbers, we needed to create a linear equation based on the condition that their sum equals \(-2\). This led us to the equation: \[ x + (3 - 2x) + (x - 6) + (2x - 11) = -2 \]A crucial feature of linear equations is combining like terms to simplify them, making them easier to solve. By gathering all the \( x \)-related terms together and separately combining all constant terms, we simplified our original equation to:\[ 2x - 14 = -2 \]Thanks to this linear equation format, finding the value of \( x \) became a simple algebraic manipulation.

Problem Solving Steps

While tackling algebraic problems, following an organized series of steps ensures a clear path to the solution. Here, we used a systematic approach to solve the given exercise:

• Define variables comprehensively to represent unknowns.
• Translate verbal conditions into algebraic expressions and create appropriate equations.
• Simplify the equation by combining like terms, which prepares it for solving.
• Isolate the variable on one side to solve the equation. For our problem, this involved adding 14 to both sides, giving \( 2x = 12 \), and then dividing each side by 2 to find \( x = 6 \).
• Substitute the value of \( x \) back into the expressions representing each number to compute their values explicitly.
• Verify the solution to ensure correctness, by checking that the sum of the computed numbers indeed equals the specified sum, \(-2\).

Following these steps, not only structures your solution but also builds a broader problem-solving skill set applicable in various mathematical contexts.