Solve the Equation: Unraveling 7a(3 + 1/a) - 2a = 0

Great question! When you multiply $7a \times \frac{1}{a}$, you can rewrite it as $\frac{7a}{a}$. Since a appears in both numerator and denominator, they cancel out (as long as $a \neq 0$), leaving you with just 7.

Break it down piece by piece! $7a(3 + \frac{1}{a})$ means multiply 7a by each term inside. So you get $7a \times 3 = 21a$ and $7a \times \frac{1}{a} = 7$. Then add them: $21a + 7$.

Variables cancel when you have the same variable in both numerator and denominator. For example, $\frac{a}{a} = 1$, $\frac{3x}{x} = 3$, or $\frac{5y^2}{y} = 5y$. Just remember the variable can't equal zero!

After distributing, you often get multiple terms with the same variable. Combining like terms simplifies the equation and makes it easier to solve. In this problem, $21a - 2a = 19a$ gives us a simpler equation to work with.

For this type of linear equation, we typically get exactly one solution. No solution occurs when you get something like $0 = 5$ (impossible). Infinite solutions happen when you get $0 = 0$ (always true).