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Chapter 1: Problem 21

Solve. $$ \frac{2}{3} x=30 $$

Short Answer

Expert verified
x = 45

Step by step solution

01

- Identify the Equation

Given the equation \[ \frac{2}{3} x = 30 \], we need to solve for the variable \(x\).
02

- Isolate the Variable

To isolate \(x\), multiply both sides of the equation by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \).
03

- Perform Multiplication

Multiply both sides: \[ x = 30 \times \frac{3}{2} \]
04

- Simplify the Right Side

Simplify the multiplication: \[ x = 30 \times \frac{3}{2} = 45 \]

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Key Concepts

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

Isolating Variables
Isolating the variable is an essential step in solving linear equations. The goal is to get the variable of interest, in this case, \(x\), by itself on one side of the equation. Let's look at our equation: \( \frac{2}{3}x = 30 \).
We need to 'undo' the fraction that is multiplying \(x\). To do that, we use the reciprocal of the fraction. This will help us eliminate the fraction and isolate \(x\).
Remember, isolating the variable makes it easier to clearly see the value it holds.
Multiplication of Fractions
When we multiply fractions, it's important to understand the steps involved. Fractions are numbers too, and obey the same rules. Let's say you have a fraction and need to multiply it.
For example, in the exercise: \( x = 30 \times \frac{3}{2} \). To multiply these:
  • Multiply the numerator (top part) with 30.
  • Then, multiply the result by the denominator (bottom part).
This simplifies to \( x = \frac{90}{2} \), which then simplifies further to \( x = 45 \).
Multiplication of fractions can seem tricky, but just approach it step-by-step.
Reciprocals
A reciprocal flips the numerator and the denominator of a fraction. This can help us solve equations easier. For instance, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
When you multiply a fraction by its reciprocal, you get 1. So \( \frac{2}{3} \times \frac{3}{2} = 1 \).
  • The key to using reciprocals is to know they 'undo' the fraction.
  • This is why multiplying both sides of \( \frac{2}{3}x = 30 \) by \( \frac{3}{2} \) isolates \(x\).
Using reciprocals can simplify many algebraic problems.

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Most popular questions from this chapter

Insert one pair of parentheses to convert each of the following false statements into a true statement. $$ 8-5^{3}+9=36 $$

Classify each statement as either true or false. The following sets are used: \(\mathrm{N}=\) the set of natural numbers; \(W=\) the set of whole numbers; \(\mathbb{Z}=\) the set of integers; \(\mathbb{C}=\) the set of rational numbers; \(\mathrm{H}=\) the set of irrational numbers; \(\mathrm{R}=\) the set of real numbers. $$ 4.3 \notin \mathbb{Z} $$

Classify each statement as either true or false. The following sets are used: \(\mathrm{N}=\) the set of natural numbers; \(W=\) the set of whole numbers; \(\mathbb{Z}=\) the set of integers; \(\mathbb{C}=\) the set of rational numbers; \(\mathrm{H}=\) the set of irrational numbers; \(\mathrm{R}=\) the set of real numbers. $$ 196 \in \mathbb{N} $$

Translate to an algebraic expression. The product of the difference of two numbers and their sum

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?. $$ -6.08,-5,0,1, \sqrt{17}, \frac{99}{2} $$
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