Solve the equation below.
$3 x-8=\frac{8 x+4}{2}$
We are given the linear equation:
\[3x - 8 = \frac{8x + 4}{2}\]
Our objective is to find the value of \(x\) that satisfies this equality. This is a linear equation involving a fraction, which we can simplify by clearing the denominator. We will aim to isolate \(x\) by gathering all terms containing \(x\) on one side and all constant numerical terms on the other.
To simplify the equation, we first multiply both sides by 2 to eliminate the fraction on the right side:
\[2 \times (3x - 8) = 2 \times \frac{8x + 4}{2}\]
Distributing the 2 on the left side, we get:
\[6x - 16 = 8x + 4\]
Now, we group the terms involving \(x\) on one side and the constant terms on the other. Subtracting \(6x\) from both sides, we obtain:
\[-16 = 2x + 4\]
Next, subtract 4 from both sides to isolate the term with \(x\):
\[-16 - 4 = 2x\]
\[-20 = 2x\]
Finally, divide by 2 to solve for \(x\):
\[x = \frac{-20}{2} = -10\]
Thus, the solution to the equation is \(x = -10\). Comparing this result with the given options, we find it matches the second option.
Linear equations are fundamental in many real-life situations. For example, if you are planning a budget, you might encounter an equation like this when trying to determine how many items you can purchase given a fixed amount of money and varying costs. By setting up an equation where one side represents your total available funds and the other side represents the sum of individual costs (like \(3x - 8\) for a certain number of units \(x\) minus a discount), you can easily solve for the unknown quantity \(x\). Mastery of this technique provides the foundation for more advanced financial and resource planning models.