What is the solution to this equation?
$3(n-9)-2(n+4)=6 n$
$n=$
The given equation is \(3(n - 9) - 2(n + 4) = 6n\). Our goal is to isolate the variable \(n\) to find its value. This is a linear equation in one variable, which we can solve by expanding the terms using the distributive property, simplifying both sides, and then rearranging the equation to solve for \(n\).
First, we apply the distributive property to expand the expressions on the left-hand side of the equation:
\[3(n - 9) = 3n - 27\]
\[-2(n + 4) = -2n - 8\]
Substituting these back into the original equation, we get:
\[(3n - 27) - (2n + 8) = 6n\]
\[3n - 27 - 2n - 8 = 6n\]
Next, we combine the like terms on the left side:
\[(3n - 2n) + (-27 - 8) = 6n\]
\[n - 35 = 6n\]
Now, we move the terms containing \(n\) to one side of the equation and the constant term to the other. Subtracting \(n\) from both sides, we obtain:
\[-35 = 6n - n\]
\[-35 = 5n\]
Finally, we solve for \(n\) by dividing both sides by 5:
\[n = \frac{-35}{5}\]
\[n = -7\]
Linear equations are essential in many everyday situations, such as budgeting or predicting costs. For example, if you are planning a party and have a base cost for renting a venue, plus a per-person fee, you can use a linear equation to determine the number of guests you can afford within a fixed budget. By representing these costs as an equation, you can solve for the unknown quantity (the number of guests) just as we solved for \(n\) in this problem.