Problem 29 - Fundamentals of Mathematics

Question

If five is subtracted from four times some number the result is negative twenty-nine. What is the number?

Step-by-Step Solution

Answer

The number is -6.

Step 1: Write the Equation

The problem statement gives us a clear relationship. Let the unknown number be represented as \( x \). According to the problem, 'four times some number' translates to \( 4x \). When we subtract five from this product, the result is negative twenty-nine. Hence, the equation can be written as: \( 4x - 5 = -29 \).

Step 2: Add Five to Both Sides

To isolate the term involving \( x \), add five to both sides of the equation: \( 4x - 5 + 5 = -29 + 5 \). This simplifies to \( 4x = -24 \).

Step 3: Solve for x

Now, we need to solve for \( x \). To do this, divide both sides by four: \( \frac{4x}{4} = \frac{-24}{4} \). Simplifying this gives \( x = -6 \).

Key Concepts

Problem Solving

Problem solving in algebra involves breaking down a word problem into manageable mathematical expressions. It requires:

• Understanding the question: Identify what is being asked, such as finding a number in this problem.
• Translating words into algebraic equations: In our example, "four times" translates to multiplying by 4.
• Setting up an equation: Subtracting five from four times a number turns into the equation \( 4x - 5 \).

This approach helps to systematically work through complex problems, simplifying the process to basic operations that can be gradually solved.

Isolation Technique

The isolation technique is a fundamental step in solving algebraic equations, especially when trying to find the value of a variable.This involves rearranging the equation so that the variable of interest, such as \( x \), stands alone on one side of the equation. Here are the steps:

• Identify the term on the left side that contains the variable, like 4\( x \), and ":-5" which needs to be "removed" first.
• Add 5 to both sides to cancel out the \(-5\) next to 4\( x \), leading us to: \( 4x = -24 \).

Once the variable term is isolated, further simplify by dividing by the coefficient of the variable. This results in adding clarity and straightforwardness to the solution process.

Linear Equations

Linear equations are equations of the first order. They involve variables that are not raised to any power other than one. The general form is \( ax + b = c \). In our exercise:

• The equation \( 4x - 5 = -29 \) is a typical linear equation.
• Such equations have a straightforward graph: a straight line, which represents all solutions.

Solving these starts with structuring the equation correctly, using isolation techniques to simplify, and finding the variable's value with basic arithmetic operations. Linear equations are common in algebra and provide foundational skills for more complex mathematics.