How do you solve the equation in detail :: 12(2w-3)=6w
The Correct Answer and Explanation is:
To solve the equation ( 12(2w – 3) = 6w ), we’ll simplify and isolate the variable ( w ). Here’s a step-by-step approach with a detailed explanation:
Step 1: Distribute 12 to terms inside the parentheses
Start by expanding the left side of the equation:
[
12 \cdot (2w – 3) = 6w
]
Using the distributive property, multiply ( 12 ) by each term inside the parentheses:
[
12 \cdot 2w – 12 \cdot 3 = 6w
]
[
24w – 36 = 6w
]
Step 2: Move all terms involving ( w ) to one side of the equation
Our goal is to isolate ( w ) on one side of the equation. To do this, move ( 6w ) from the right side to the left side by subtracting ( 6w ) from both sides:
[
24w – 6w – 36 = 0
]
Simplify by combining like terms on the left side:
[
18w – 36 = 0
]
Step 3: Isolate the term with ( w )
Now, add ( 36 ) to both sides to move the constant term to the right:
[
18w = 36
]
Step 4: Solve for ( w )
Now that we have ( 18w = 36 ), we can solve for ( w ) by dividing both sides by ( 18 ):
[
w = \frac{36}{18}
]
[
w = 2
]
Final Answer
The solution to the equation is ( w = 2 ).
Detailed Explanation
This solution involves applying fundamental algebraic principles, particularly the distributive property, combining like terms, and isolating the variable. First, distributing ( 12 ) across the terms in the parentheses simplifies the left side of the equation. This step is crucial because it eliminates the parentheses, allowing us to deal directly with terms involving ( w ).
Next, by moving all terms involving ( w ) to one side, we ensure that all ( w )-terms are consolidated, making it easier to isolate ( w ) by combining like terms. The final steps involve using addition and division to isolate ( w ), a process known as “solving for ( w )” because it leaves ( w ) alone on one side of the equation.
Verifying the solution by substituting ( w = 2 ) back into the original equation confirms that both sides are equal, showing the correctness of the answer. Thus, the solution to ( 12(2w – 3) = 6w ) is ( w = 2 ).