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Chapter 3: Problem 27

Solve the equation. $$30 b=5$$

Short Answer

Expert verified
The solution to the equation is \( b = \frac{1}{6} \)

Step by step solution

01

Identify the type of the equation

The given equation is of the form \(ab = c\),where 'a' is the coefficient, 'b' is the variable and 'c' is the constant.
02

Isolate the variable

The solution involves isolating \(b\). To do this, we need to cancel out 30 from the left side of the equation. To achieve this, we can divide both sides of the equation by 30: \(\frac{30b}{30} = \frac{5}{30}\).
03

Simplify the equation

After simplifying, the left side becomes \(b = \frac{5}{30}\).

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Key Concepts

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

Isolate the Variable
In algebra, solving an equation typically involves isolating the variable. This means rearranging the equation until the variable of interest stands alone on one side of the equation. For our given equation, \(30b = 5\), we need to get \(b\) by itself.
  • First, identify the operations surrounding the variable. Here, \(b\) is being multiplied by 30.
  • To isolate \(b\), perform the inverse operation; since 30 is multiplication, the inverse is division.
  • Divide both sides of the equation by 30 to cancel it out from the left side.
This strategic process of isolating the variable is fundamental. It simplifies the equation and lays the foundation for solving for the unknown. It is also a key step in nearly all algebraic manipulations.
Simplifying Equations
Once you have begun to isolate the variable, the next crucial step is to simplify the equation. For the equation \(\frac{30b}{30} = \frac{5}{30}\), simplification now involves reducing both sides to their simplest form. Here's how it works:
  • The left side simplifies straightforwardly because \(\frac{30b}{30} = b\), as the 30s cancel each other out.
  • The right side requires performing the division \(\frac{5}{30}\). This fraction simplifies by dividing both numerator and denominator by their greatest common divisor, which is 5.
  • So, \(\frac{5}{30}\) simplifies to \(\frac{1}{6}\).
Simplifying is essential for obtaining the most understandable and actionable form of an equation. It equips students with a clear path to find solutions and verify their accuracy.
Division in Algebra
Division in algebra is an operation that reduces or distributes values evenly. It is especially useful in solving equations where you need to balance and simplify expressions. When working with the equation \(30b = 5\), division helps isolate the variable:
  • Understand that division is the inverse of multiplication, which allows the multiplication factor alongside the variable to be canceled.
  • Divide both sides of the equation simultaneously to maintain equality. This means dividing each term by the same non-zero number, 30, in this case.
  • After division, the equation becomes \(b = \frac{1}{6}\). This demonstrates the isolated variable with a clear value.
Division is a fundamental operation in algebra, integral for solving equations by reducing coefficients and forming isolated expressions. It paves the way for accurately finding values of variables in various mathematical situations.

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