Select the correct answer.
Which value of $n$ makes the equation true?
$6 n-2(4 n+7)+4=-2$
A. -6
B. -4
C. 4
D. 6
Answer (1)
Distribute the -2: 6 n − 8 n − 14 + 4 = − 2 .
Combine like terms: − 2 n − 10 = − 2 .
Isolate the variable: − 2 n = 8 .
Solve for n : n = − 4 . The correct answer is − 4 .
Explanation
Problem Analysis We are given the equation 6 n − 2 ( 4 n + 7 ) + 4 = − 2 and asked to find the value of n that makes the equation true. We will solve this equation by expanding, combining like terms, and isolating n .
Expanding the Equation First, distribute the − 2 to the terms inside the parentheses: 6 n − 2 ( 4 n ) − 2 ( 7 ) + 4 = − 2
6 n − 8 n − 14 + 4 = − 2
Combining Like Terms Next, combine like terms on the left side of the equation: ( 6 n − 8 n ) + ( − 14 + 4 ) = − 2 − 2 n − 10 = − 2
Isolating the Variable Now, isolate the term with n by adding 10 to both sides of the equation: − 2 n − 10 + 10 = − 2 + 10 − 2 n = 8
Solving for n Finally, solve for n by dividing both sides of the equation by − 2 :
− 2 − 2 n = − 2 8 n = − 4
Checking the Solution To check our solution, substitute n = − 4 back into the original equation: 6 ( − 4 ) − 2 ( 4 ( − 4 ) + 7 ) + 4 = − 2 − 24 − 2 ( − 16 + 7 ) + 4 = − 2 − 24 − 2 ( − 9 ) + 4 = − 2 − 24 + 18 + 4 = − 2 − 6 + 4 = − 2 − 2 = − 2
The solution checks out.
Final Answer Therefore, the value of n that makes the equation true is n = − 4 .
Examples
In electrical engineering, this type of equation can be used to determine the value of a resistor in a circuit given certain voltage and current constraints. By setting up an equation that models the circuit's behavior, we can solve for the unknown resistance value, ensuring the circuit operates as intended. For example, if the equation represents the voltage drop across a series of resistors, solving for 'n' (representing resistance) helps in designing circuits with specific characteristics.
