To determine how many solutions the equation \(15x - \frac{5}{3} = 5x + 4\) has, we can solve for \(x\).
First, let's isolate \(x\) by rearranging the equation. We'll start by moving all terms involving \(x\) to one side and constant terms to the other:
-
Subtract \(5x\) from both sides:
\[ 15x - 5x - \frac{5}{3} = 4 \]
This simplifies to:
\[ 10x - \frac{5}{3} = 4 \]
-
Next, add \(\frac{5}{3}\) to both sides:
\[ 10x = 4 + \frac{5}{3} \]
To combine \(4\) and \(\frac{5}{3}\), we convert \(4\) to a fraction with a denominator of \(3\):
\[ 4 = \frac{12}{3} \]
So,
\[ 10x = \frac{12}{3} + \frac{5}{3} = \frac{12 + 5}{3} = \frac{17}{3} \]
-
Now, divide both sides by \(10\) to solve for \(x\):
\[ x = \frac{17}{3} \cdot \frac{1}{10} = \frac{17}{30} \]
Since we found exactly one value of \(x\), the equation \(15x - \frac{5}{3} = 5x + 4\) has exactly one solution.
Thus, the answer is:
B) exactly one solution