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Chapter 2: Problem 66

For exercises 1-72, (a) solve. (b) check. $$ 0=9(b+8)-12(b+2) $$

Short Answer

Expert verified
The solution is \( b = 16 \).

Step by step solution

01

- Distribute the constants

Expand the equation by distributing the constants inside the parentheses: \[ 0 = 9(b + 8) - 12(b + 2) \] \[ 0 = 9b + 72 - 12b - 24 \]
02

- Combine like terms

Combine the terms involving \(b\) and the constant terms: \[ 0 = 9b - 12b + 72 - 24 \] \[ 0 = -3b + 48 \]
03

- Solve for \(b\)

Isolate \(b\) by moving the constant to the other side of the equation: \[ -48 = -3b \] Divide both sides by -3: \[ b = 16 \]
04

- Check the solution

Substitute \( b = 16 \) back into the original equation to verify: \[ 0 = 9(16 + 8) - 12(16 + 2) \] \[ 0 = 9(24) - 12(18) \] \[ 0 = 216 - 216 \] Since both sides are equal, the solution \( b = 16 \) is correct.

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Key Concepts

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

Distributive Property
The distributive property is a fundamental algebraic property used to simplify expressions. It states that: a(b + c) = ab + ac.
In the given exercise, we need to apply the distributive property to both terms involving parentheses. Here's how it works:
Step 1: Expand

  • For the first term: 9(b + 8) = 9b + 72
  • For the second term: -12(b + 2) = -12b - 24
Applying this property makes the equation simpler and easier to handle, transforming it into: 0 = 9b + 72 - 12b - 24.
The next step involves combining like terms, another crucial algebra concept.
Combining Like Terms
Combining like terms means grouping together terms with the same variable to simplify the equation. In our example, we have two types of terms: those involving the variable b and constant terms:

Step 2: Combine

  • Combine the b terms: 9b - 12b which gives -3b

  • Combine the constants: 72 - 24 which gives 48

This simplifies the equation to: 0 = -3b + 48.
With the equation now simplified, we are ready to solve for the variable b.
Isolating Variables
Isolating the variable is the process of manipulating the equation to get the variable on one side and everything else on the other side. Let's solve for b:

Step 3: Solve
Move the constant term to the other side of the equation, aiming to isolate b:

  • Start with: 0 = -3b + 48

  • Subtract 48 from both sides to get: -48 = -3b

Now, to isolate b, divide both sides by -3:

  • Divide: -48 / -3 = b

  • Simplify to find: b = 16
With b isolated, we have found the solution: b = 16. The final step is to verify if this solution is correct.
Verification of Solution
Verification means substituting the solution back into the original equation to check if it satisfies it. Here's how we do it:

Step 4: Verify

  • Substitute b = 16 into the original equation: 0 = 9(16 + 8) - 12(16 + 2)

  • Simplify inside the parentheses: 9(24) - 12(18)

  • Calculate: 216 - 216 = 0
Since both sides of the equation are equal, b = 16 is verified as the correct solution.
This step ensures that our method and calculations are accurate, providing confidence in the solution.

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Most popular questions from this chapter

For exercises 93-96, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(-2 x+12>36\) Incorrect Answer: \(-2 x+12>36\) $$ \begin{aligned} -12 &-12 \\ \hline-2 x+0 &>24 \\ \frac{-2 x}{-2} &>\frac{24}{-2} \\ x &>-12 \end{aligned} $$

The rectangular playing field of a pool table is 50 in. wide and 100 in. long. The top of the entire table, including the playing field and the area of the table that surrounds the playing field, is 62 in. wide and 112 in. long. Find the area of the table that surrounds the playing field.

Solve \(V=\frac{L W T}{144}\) for \(W\).

For exercises 9-36, (a) solve. (b) check the direction of the inequality sign. $$ 4 a-12<-60 $$

For exercises 89-92, solve. Use a calculator to do the arithmetic. $$ -\frac{3}{8}=\frac{b}{160} $$
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