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Chapter 3: Problem 13

Solve the equations. $$ B-4(B-3(1-B))=57 $$

Short Answer

Expert verified
Answer: B = -3.

Step by step solution

01

Distribute the terms inside the parentheses

Firstly, distribute the -4 to both of the terms inside the parentheses, which results in: $$ B - 4(B - 3(1 - B)) = 57 $$ becomes $$ B - 4(B - 3 + 3B) = 57 $$ Now, distribute the -4 to both (B-3) and the (3B): $$ B - 4(B - 3) - 4(3B) = 57 $$ Which simplifies to: $$ B - 4B +12 -12B = 57 $$
02

Combine like terms and solve for B

Combine all the B terms (B, -4B, and -12B) and move the constant (12) to the other side of the equation: $$ B - 4B - 12B = 57 - 12 $$ This simplifies to: $$ -15B = 45 $$ Now, to isolate B, divide both sides of the equation by -15: $$ B = \frac{45}{-15} $$
03

Calculate the final value for B

Divide 45 by -15 to find the value for B: $$ B = -3 $$ So, the solution to the equation is B = -3.

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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 one of the foundational concepts in algebra that allows you to simplify complex expressions. It states that when a number is multiplied by a sum or difference inside a set of parentheses, you can "distribute" the multiplication to each term inside the parentheses. For instance, given an expression like \(-4(B - 3(1-B))\), you distribute the \(-4\) to each term inside.
To break it down:
  • First, simplify the innermost expression \(3(1-B)\) which becomes \(3 - 3B\).
  • Now replace the simplified expression and continue, \(-4(B - 3 + 3B)\), resulting in \(-4(B - 3) - 4(3B)\).
This process is crucial because it transforms complex equations into more manageable ones by distributing and reorganizing the terms.
Combining Like Terms
Combining like terms is a vital step for simplifying algebraic expressions and solving equations. Like terms are terms that contain the same variable raised to the same power.
In the equation you are solving, \(B - 4B + 12 - 12B = 57\), you can identify similar terms based on their variable part.
  • Combine all the \(B\) terms: \(B - 4B - 12B\), which simplifies to \(-15B\).
  • The number \(12\) is a constant and thus handled separately by moving it to the other side of the equation, giving you \(57 - 12\).
Bringing like terms together simplifies the equation, paving the way for a more direct solution.
Solving Linear Equations
Solving linear equations entails finding the value of the unknown variable that makes the equation true. After applying the distributive property and combining like terms, the equation becomes \(-15B = 45\).
The next step is to isolate the variable \(B\). This is done by performing the inverse operation on both sides of the equation.
  • Since \(B\) is being multiplied by \(-15\), divide both sides by \(-15\) to get \(B = \frac{45}{-15}\).
  • Calculate the result: dividing 45 by \(-15\) yields \(B = -3\).
The solution \(B = -3\) satisfies the equation, illustrating a systematic approach to finding the value of \(B\) that solves the equation. Linear equations like this one often require similar strategies.

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