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Chapter 3: Problem 11

solve the equation using the multiplication or division properties of equality. $$14 b=42$$

Short Answer

Expert verified
The solution is \( b = 3 \).

Step by step solution

01

Identify the Variable

In the equation, the variable to solve for is \(b\).
02

Use the Division Property of Equality

To isolate \(b\), divide both sides of the equation by 14: \[ \frac{14b}{14} = \frac{42}{14} \]
03

Simplify the Equation

Simplify both sides of the equation: \[ b = 3 \]

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Key Concepts

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

Division Property of Equality
The Division Property of Equality helps maintain the balance of an equation when you divide both sides by the same non-zero number.
This means if you have an equation like \(14b = 42\), you can divide both sides by 14.
This step keeps the equation true and is crucial in solving for the variable.
\[ \frac{14b}{14} = \frac{42}{14} \]
After dividing, you get: \[ b = 3 \]
By understanding this property, you can confidently manage and solve equations.
Just remember: dividing by zero is not allowed as it breaks the rules of arithmetic.
Isolating the Variable
Isolating the variable means getting the unknown value alone on one side of the equation.
In the equation \(14b = 42\), we aim to isolate \(b\).
Using the Division Property of Equality, we divide both sides by 14, simplifying the equation to just \(b\).
So, we have: \[ b = \frac{42}{14} \]
This simplifies to: \[ b = 3 \]
Isolating the variable makes it easier to see the solution clearly.
Always perform the same operation on both sides to keep the equation balanced.
Simplifying Equations
Simplifying equations involves reducing expressions to their simplest form.
After isolating \(b\) in the equation \(14b = 42\) by dividing both sides by 14, we get: \[ b = \frac{42}{14} \]
Now, you simplify \( \frac{42}{14} \) to its smallest form, \[ 42 \div 14 = 3 \]
This gives us: \[ b = 3 \]
Simplified equations are easier to handle and understand.
Reducing expressions helps you quickly see the solution.

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