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Chapter 6: Problem 5

Determine whether the expressions are equal. $$\frac{x}{3}, \frac{3 x}{9}$$

Short Answer

Expert verified
Yes, the expressions \( \frac{x}{3} \) and \( \frac{3 x}{9} \) are equal.

Step by step solution

01

Simplify each expression

We begin by simplifying each fraction, if it can be simplified. The first expression is \( \frac{x}{3} \), which can't be simplified any further since there isn't a factor common to x and 3. The second expression is \( \frac{3x}{9} \). We can simplify this by dividing both the numerator and denominator by 3 which gives us \( \frac{x}{3} \).
02

Compare Expressions

After simplification, both expressions turn out to be \( \frac{x}{3} \). This means the two expressions are equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Simplification
Fraction simplification is the process of altering a fraction to its simplest form. This involves reducing the numerator and the denominator to the lowest possible whole numbers while maintaining their original ratio.

Here's how it works:
  • Identify common factors: Look for numbers that divide both the numerator and the denominator evenly.
  • Divide by the greatest common factor (GCF): Use this number to simplify the fraction.
In the provided step by step solution, you simplify the fraction \( \frac{3x}{9} \). This is done by dividing both the numerator (3x) and the denominator (9) by their GCF, which is 3. After simplification, the fraction becomes \( \frac{x}{3} \).

This simplification helps in easily comparing, adding, or solving problems involving fractions.
Equivalent Expressions
Equivalent expressions arise when two expressions yield the same result, despite possibly being written differently.

This concept can become clearer when fractions are involved. Upon simplifying different-looking fractions, you might find they are equivalent. For instance:
  • The original expressions \( \frac{x}{3} \) and \( \frac{3x}{9} \) seem different but simplify to \( \frac{x}{3} \).
Because these match exactly after simplification, they are classified as equivalent expressions. Recognizing equivalent expressions is crucial, as they can help us solve equations more easily and verify if expressions are equal. It鈥檚 an essential practice for working effectively with algebraic expressions and always helps in checking your work.
Algebraic Equality
Algebraic equality involves determining if two expressions represent the same value. This forms the foundation of solving equations because it tells us if two statements hold the same mathematical truth.

In the given exercise, after simplifying both fractions, we ended with two identical expressions \( \frac{x}{3} \). This demonstrates algebraic equality. Essentially, algebraic equality requires comparing two expressions.
  • If they simplify to the same result, they are equal.
This concept helps solve many algebraic problems, such as validating identities or balancing equations. Understanding how to establish and confirm algebraic equality is a skill that empowers students to tackle complex math challenges with confidence.

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Most popular questions from this chapter

Refer to the tax deductions listed in the table. Give each ratio in simplest form. $$\begin{array}{lc}\text { Item } & \text { Amount } \\\\\hline \text { Medical } & \$ 995 \\\\\text { Real estate tax } & \$ 1,245 \\\\\text { Contributions } & \$ 1,680 \\\\\text { Mortgage interest } & \$ 4,580 \\\\\text { Union dues } & \$ 225\end{array}$$ Find the ratio of real estate tax deductions to the total deductions.

Let \(x\) equal a number of your choosing. Without simplifying first, use a calculator to evaluate $$\frac{x^{2}+x-6}{x^{2}+3 x} \cdot \frac{x^{2}}{x-2}$$ Try again, with a different value of \(x\). If you were to simplify the expression, what do you think you would get?

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{6 a-6 b+6 c}{9 a-9 b+9 c}$$

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{2 x^{2}-8}{x^{2}-3 x+2}$$

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{3 x^{2}-27}{x^{2}+3 x-18}$$
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