Solve for $n$. $11(n-1)+35=3n$
Answer (1)
Expand the left side of the equation: 11 ( n − 1 ) + 35 = 11 n − 11 + 35 .
Simplify the equation: 11 n + 24 = 3 n .
Isolate n : 8 n = − 24 .
Solve for n : n = − 3 . The final answer is − 3 .
Explanation
Understanding the Problem We are given the equation 11 ( n − 1 ) + 35 = 3 n and asked to solve for n . We will expand the left side, simplify, and isolate n to find the solution.
Expanding the Left Side First, we expand the left side of the equation: 11 ( n − 1 ) + 35 = 11 n − 11 + 35
Simplifying the Equation Next, we simplify the left side by combining the constant terms: 11 n − 11 + 35 = 11 n + 24
Setting up the Equation Now, we set the simplified left side equal to the right side: 11 n + 24 = 3 n
Isolating n To isolate n , we subtract 3 n from both sides of the equation: 11 n − 3 n + 24 = 3 n − 3 n
8 n + 24 = 0
Further Isolating n Next, we subtract 24 from both sides: 8 n + 24 − 24 = 0 − 24 8 n = − 24
Solving for n Finally, we divide both sides by 8 to solve for n :
8 8 n = 8 − 24 n = − 3
Verification To verify our solution, we substitute n = − 3 back into the original equation: 11 ( − 3 − 1 ) + 35 = 3 ( − 3 ) 11 ( − 4 ) + 35 = − 9 − 44 + 35 = − 9 − 9 = − 9 Since the equation holds true, our solution is correct.
Final Answer Therefore, the solution to the equation 11 ( n − 1 ) + 35 = 3 n is n = − 3 .
Examples
Imagine you're managing a small business and need to calculate your profit. You have a fixed cost of $11 per item, but you get a bulk discount that reduces the total cost by $35. If you sell each item for $3, you can use this equation to determine how many items you need to sell to break even. Solving for 'n' tells you the number of items needed to make your costs equal your revenue, which is a fundamental concept in business and economics.
