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Chapter 3: Problem 74

Solve the equation. $$-(b-11)-3 b-11=-16$$

Short Answer

Expert verified
b = 4

Step by step solution

01

Distribute the negative sign

First, distribute the negative sign through the parentheses: \( -(b-11) = -b+11 \). So the equation becomes: \( -b + 11 - 3b - 11 = -16 \).
02

Combine like terms

Next, combine the like terms on the left side of the equation: \( -b - 3b + 11 - 11 = -16 \). Simplify it to: \( -4b = -16 \).
03

Solve for b

Finally, solve for \( b \) by dividing both sides of the equation by -4: \( b = \frac{-16}{-4} \). So, \( b = 4 \).

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Key Concepts

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

Distributing Negative Signs
When solving linear equations, it's crucial to handle negative signs correctly. In our equation, we have a negative sign outside the parentheses, \( -(b-11) \). To simplify this, we need to distribute the negative sign to both terms inside the parentheses:
\ (-(b-11) = -b + 11) \.
This changes the sign of both terms. The \( b \) becomes \( -b \) and \( -11 \) becomes \( +11 \).
This step ensures that our equation reflects the correct values as we proceed.
Combining Like Terms
After distributing negative signs, the equation is \( -b + 11 - 3b - 11 = -16 \).
Now, it's time to simplify by combining like terms. Like terms are terms that contain the same variable raised to the same power. In our equation, \ -b \ and \ -3b \ are like terms because they both contain the variable \ b \.
Combine these terms: \ -b - 3b = -4b \.
Notice the constants, \ 11 \ and \ -11 \, which cancel each other out: 11 - 11 = 0.
So, the equation simplifies to \ -4b = -16 \.
Isolating the Variable
The final step is to isolate the variable \ b \, so we can find its value. Our simplified equation is \ -4b = -16 \.
To isolate \ b \, we need to divide both sides of the equation by \ -4 \.
\[ b = \frac{-16}{-4} \]
Solving this, we get \ b = 4 \.
This means that \ b \ must be 4 for the original equation to be true.

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