Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the given information to write an equation. Let \(x\) represent the number described in each exercise. Then solve the equation and find the number. If a number is multiplied by \(-6,\) the result is \(20 .\) Find the number.

Step-by-Step Solution

Answer

The number is 15.

Step 1: Formulation of Equation

Begin by translating the given information into an algebraic equation. The problem states that if 23 is subtracted from a certain number, the result is -8. We can express this with \(x\) representing the unknown number, giving us the equation: \(x - 23 = -8\).

Step 2: Solving the Equation

To isolate \(x\) and solve for it, add 23 to both sides of the equation. This gives: \(x = -8 + 23\).

Step 3: Finding the Number

Finally, perform the operation on the right hand side to find the value of \(x\). This results in: \(x = 15\).

Key Concepts

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. In solving equations, our goal is to find the values of the unknowns that make the equation true. Let's walk through the process using the example: \(x - 23 = -8\). Solving equations generally involves a few logical steps:

• Identify the equation that needs to be solved.
• Use appropriate operations to isolate the unknown variable on one side of the equation.
• Perform arithmetic operations to simplify the equation.

In our example, we start with the equation \(x - 23 = -8\). To isolate \(x\), we perform the opposite operation of subtraction, which is addition. By adding 23 to both sides of the equation, we maintain balance and work towards isolating \(x\).

Adding 23 gives us: \(x = -8 + 23\). The calculations are simplified, leading us to the solution \(x = 15\). Always verify your solution by plugging it back into the original equation to ensure it satisfies all given conditions.

Unknown Variables

Variables play a crucial role in algebra. They are symbols used to represent unknown values or numbers. In our exercise, \(x\) is the unknown variable representing the number we're trying to find. Understanding the concept of variables is fundamental in translating real-world scenarios into algebraic expressions and equations.

Variables allow for flexibility and generalization in mathematics:

• They make it possible to express problems in a general way, applicable to various situations.
• Variables stand in place of unknowns until we use mathematical methods to solve for them.

In the equation \(x - 23 = -8\), \(x\) is the unknown number. By performing operations on both sides of the equation, we solve for \(x\). This technique of isolating variables is what helps us move from a statement with unknowns to finding specific numeric answers.

The clarity and logic that come from understanding variables are indispensable tools in algebra and beyond.

Integer Operations

Integer operations are basic but essential arithmetic actions involving whole numbers both positive and negative. In the context of solving equations like \(x - 23 = -8\), understanding integer operations helps simplify and solve equations accurately.

Let's consider integer addition as it applies to our exercise:

• Addition of a negative integer is equivalent to subtraction of its positive counterpart (e.g., adding -8 is the same as subtracting 8).
• Addition of a series of integers follows the associative and commutative laws, allowing flexible grouping and reordering of numbers.

In our solution, when we add 23 to both sides of the equation to isolate \(x\), we're performing an integer operation by combining negative and positive integers: \(-8 + 23\).

This operation requires understanding that moving along the number line, adding a larger positive number to a smaller negative one results in a positive number, hence \(x = 15\). Recognizing the properties of integers simplifies the challenge of solving algebraic equations by enabling accurate computational steps.