Problem 26 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using the multiplication property of equality. Be sure to check your proposed solutions. $$-\frac{x}{5}=-1$$

Step-by-Step Solution

Answer

The solution to the equation \(-\frac{x}{5} =-1\) is x=5

Step 1: Apply the multiplication property of equality

Multiply both sides of the equation \(-\frac{x}{5}=-1\) by -5. This gives: \[ -5 \cdot -\frac{x}{5} = -1 \cdot -5 \] which simplifies to: \[x = 5\]

Step 2: Check the solution

Substitute x=5 back into the original equation to verify that it satisfies the equation: \[-\frac{5}{5} = -1\] which simplifies to \[-1=-1\] Therefore, x=5 is indeed a solution to the original equation.

Key Concepts

Solving Equations

When you're faced with an equation like \(-\frac{x}{5}=-1\), the goal is to find the value of \(x\) that makes the equation true. One effective tool for this is the multiplication property of equality. This property tells us that if we multiply both sides of an equation by the same number, the equation remains balanced.

In our example, we need to get \(x\) by itself on one side of the equation. To do this, we multiply both sides by \(-5\):

• Left side: \(-5 \cdot -\frac{x}{5}\)
• Right side: \(-1 \cdot -5\)

This operation knocks out the fraction, leaving us with a simpler form: \(x = 5\).

Using the multiplication property of equality helps in solving many types of equations, gradually helping you untangle complex problems with greater ease.

Checking Solutions

After you've found a potential solution, it's crucial to check your work. Verifying the solution ensures you didn't make any miscalculations and that your answer actually satisfies the original equation.

To confirm you're on the right track, substitute the value back into the original equation. In our example, substitute \(x = 5\):

• \(-\frac{5}{5} = -1\)

Simplifying gives us \(-1 = -1\), showing that both sides of the equation match perfectly.

By checking solutions, you not only confirm your answer is correct, but you also strengthen your understanding of the process for solving equations. This step is like double-checking your homework – it gives you confidence and ensures precision.

Algebra

Algebra is a foundational branch of mathematics where we often deal with variables and operations. It introduces tools and symbols that help solve problems systematically. Understanding algebra enables you to manipulate equations and find unknown values, which is a key skill in both academics and real-world applications.

In the example equation \(-\frac{x}{5}=-1\), algebra teaches us how to shift variables and constants efficiently using operations like addition, subtraction, multiplication, and division. The multiplication property of equality is just one technique in the algebra toolbox that helps simplify and solve equations.

Engaging with algebra opens the door to advanced math topics and enhances logical thinking. It's not just about solving for \(x\); it's also about developing a systematic approach to understanding and solving all kinds of problems.