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Chapter 2: Problem 6

For exercises 1-10, solve. $$ \frac{x}{6}=30 $$

Short Answer

Expert verified
x = 180

Step by step solution

01

Understand the equation

Given the equation \(\frac{x}{6}=30\). This is a simple algebraic equation where the variable is divided by 6.
02

Isolate the variable

To solve for \(x\), multiply both sides of the equation by 6 to cancel out the division by 6 on the left side: \(\frac{x}{6} \times 6 = 30 \times 6\).
03

Simplify the equation

Simplify the result of both sides to find the value of \(x\): \(x = 180\).

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Key Concepts

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

Isolate the Variable
When solving algebraic equations, one of the most crucial steps is to isolate the variable. This means you need to get the variable by itself on one side of the equation.

In the exercise, we start with the equation \(\frac{x}{6} = 30\). Here, the variable \(x\) is divided by 6. To isolate \(x\), we need to undo the division by performing the opposite operation, which is multiplication.

We multiply both sides of the equation by 6:
\[ \frac{x}{6} \times 6 = 30 \times 6 \]

Now, the variable \(x\) is by itself on the left side. Isolating the variable is a foundational concept in algebra because it allows you to find the value of the unknown.
Algebraic Simplification
Algebraic simplification involves combining like terms and reducing expressions to their simplest form. This makes equations easier to work with.

In our step-by-step solution, we simplify the expression after isolating the variable:
\[ \frac{x}{6} \times 6 = 30 \times 6 \]

Simplifying both sides, we get \(x = 180\).

Simplification might also involve canceling terms, combining like terms, or factoring expressions depending on the complexity of the equation. It's an essential skill for handling more complicated algebraic problems.
Multiplication in Algebra
Multiplication in algebra is used to scale numbers and variables. It is also used to undo division when isolating variables.

In the original exercise, \(x\) is divided by 6. We use multiplication to cancel out this division:
\[ \frac{x}{6} \times 6 = 30 \times 6 \]

This step relies on the understanding that multiplying a number by its reciprocal should yield 1. Hence, \( \frac{x}{6} \times 6 = x \).

Understanding how to use multiplication to manipulate equations will help you solve a wide range of algebraic problems.

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Most popular questions from this chapter

Each of the three sides of an equilateral triangle is the same length. If \(P\) is the perimeter of the triangle and \(L\) is the length of a side, write a formula in \(P\) and \(L\) for the perimeter of an equilateral triangle and solve this formula for \(L\).

For exercises 99-102, solve the equation. Use a calculator to do arithmetic. $$ 21,427=815(4 d-9) $$

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For exercises 11-46, (a) solve. (b) check. $$ \frac{4}{7}=\frac{u}{84} $$

Describe the difference in the solution of \(x<9\) and the solution of \(x \leq 9\).
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