Solve $\frac{4}{5} k+14=-\frac{11}{5} k-4$.
We are given the linear equation \(\frac{4}{5}k + 14 = -\frac{11}{5}k - 4\). The variable to solve for is \(k\). Our goal is to isolate \(k\) on one side of the equation to find its value. We notice that the equation contains fractional coefficients, so we will combine like terms involving \(k\) first, then isolate the constant term.
To gather all the terms containing \(k\) on the left side of the equation, we add \(\frac{11}{5}k\) to both sides:
\[\frac{4}{5}k + \frac{11}{5}k + 14 = -4\]
Since the denominators are the same, we can add the numerators:
\[\frac{4+11}{5}k + 14 = -4\]
\[\frac{15}{5}k + 14 = -4\]
Simplifying the fraction \(\frac{15}{5}\), we get \(3\):
\[3k + 14 = -4\]
Now we isolate the term with \(k\) by subtracting \(14\) from both sides of the equation:
\[3k = -4 - 14\]
\[3k = -18\]
Finally, we solve for \(k\) by dividing both sides of the equation by \(3\):
\[k = \frac{-18}{3}\]
\[k = -6\]
Thus, the solution to the equation is \(-6\).
Linear equations are essential in everyday budgeting and resource management. For example, if you are comparing two different monthly subscription plans for a service, one might have a fixed startup fee with a low monthly cost, while the other has no startup fee but a higher monthly cost. By setting up a linear equation where the costs are equal, you can find the exact number of months for which the total cost for both services will be the same, helping you make an informed financial decision.