/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Solve the equation. $$\frac{b}... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 19

Solve the equation. $$\frac{b}{3}-12=-9$$

Short Answer

Expert verified
b = 9

Step by step solution

01

Isolate the Variable Term

First, remove the constant term on the left side by adding 12 to both sides of the equation: \(\frac{b}{3} - 12 + 12 = -9 + 12\) This simplifies to \(\frac{b}{3} = 3\)
02

Solve for the Variable

To solve for b, multiply both sides of the equation by 3: \(\frac{b}{3} \times 3 = 3 \times 3\) This simplifies to \(b = 9\)

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Key Concepts

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

Isolating Variables
Solving equations often requires us to isolate the variable. This means getting the variable (like 'b' in our given problem) by itself on one side of the equation. The goal is to make it look like 'variable = number.'
In our example, we started with \(\frac{b}{3} - 12 = -9\). The first step was to get rid of the '-12' term next to \(\frac{b}{3}\). To do that, we added 12 to both sides. This step ensures the variable term is all by itself:
\(\frac{b}{3} - 12 + 12 = -9 + 12\).
After simplifying, we get \(\frac{b}{3} = 3\). Now, the equation is simpler, and the variable term \(\frac{b}{3}\) is isolated.
Addition Property of Equality
The addition property of equality states that if you add the same number to both sides of an equation, the equality remains true. This property is essential for isolating variables.
Let's consider the given equation again: \(\frac{b}{3} - 12 = -9\).
To isolate \(\frac{b}{3}\), we had to eliminate the '-12.' We did this by adding 12 to both sides, like so: \(\frac{b}{3} - 12 + 12 = -9 + 12\).
Simplifying both sides, we got \(\frac{b}{3} = 3\). Notice how adding 12 to both sides maintained the balance of the equation. This approach is consistent with the addition property of equality and helps us in isolating variables efficiently.
Multiplication Property of Equality
The multiplication property of equality tells us that if we multiply both sides of an equation by the same number, the equality holds, and the result is a new equation that's still true.
Once we had the simplified equation: \(\frac{b}{3} = 3\), the next step was to get rid of the fraction. We did this by multiplying both sides by 3. This step looked like:
\(\frac{b}{3} \times 3 = 3 \times 3\).
Multiplying both sides by 3 simplifies the left side to 'b,' giving us: \b = 9\. This method is particularly useful for dealing with fractions or any division in equations. By applying the multiplication property of equality, we turn the equation into something very straightforward.

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Most popular questions from this chapter

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