Problem 62 - Introductory Algebra for College Students

Question

In 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 divided by \(-7,\) the result is 8. Find the number.

Step-by-Step Solution

Answer

The number sought, represented by x, is -56.

Step 1: Set up the equation

Based on the problem, we can set up the equation like this: \(x/-7=8\). This equation is derived from what the problem states: the number (x), divided by -7, equals 8.

Step 2: Solve the equation

The next step is to solve the equation to find x. Multiply both sides of the equation by -7 to isolate x: \(-7*(x/-7) = -7*8\). This simplifies the equation to \(x = -56\).

Step 3: Verify the solution

A good way to confirm the solution is to substitute x with -56 into the original equation: \(-56/-7 == 8\). Since -56 divided by -7 equals 8, our solution is correct.

Key Concepts

Equation Setup

In algebra, setting up the equation is like defining the problem structure. Essentially, you convert a word problem into a mathematical equation. It's the first and crucial step in solving any algebraic problem. Let's break it down using the given scenario.

The problem states that a number divided by -7 equals 8. We need to express this in algebraic terms. Here's how we'll do that:

• We use the variable \(x\) to represent the unknown number.
• The division by -7 is expressed as \(x / -7\).
• The equation is completed by setting it equal to 8.

Thus, the setup yields the equation: \(\frac{x}{-7} = 8\). Establishing this equation from the statement given is foundational to solving the problem. It's about translating English into math.

Multiplication by Inverse

Once our equation is set up, the next step involves isolating the unknown, \(x\). Multiplication by the inverse is a powerful mathematical tool used for this purpose. Let's understand this better.

In the equation \(\frac{x}{-7} = 8\), our goal is to solve for \(x\). The term \(-7\) here is dividing \(x\), so we use the inverse operation, which is multiplication, to remove \(-7\) from the denominator.

• The inverse of division is multiplication. Thus, we multiply both sides by \(-7\).
• On the left side: \(-7 \times \frac{x}{-7} = x\) because the \(-7\)s cancel each other out.
• On the right side: \(-7 \times 8 = -56\) completes the calculation.

This step successfully isolates \(x\) by leveraging the property that multiplying by the inverse of a number cancels it out, revealing \(x = -56\).

Division Properties

Understanding the properties of division is crucial for confirming our solutions and ensuring accuracy. Division properties help bring clarity to operations involving fractions and negative numbers.

When we verify our solution, we check if everything sums up correctly. For our derived solution \(x = -56\), we substitute back into the original equation to ensure correctness: \(-56/-7 = 8\).

• Naturally, dividing \(-56\) by \(-7\) flows by using the rule: dividing a negative by a negative yields a positive.
• Here, the actual division \(-56 \div -7\) results in \(8\), matching our expected value.

Mastering division properties helps with not just solving equations but also verifying steps, thus ensuring a thorough understanding of the concepts involved. The consistency of division across different number types consolidates our algebraic solutions.