Solve the equation with variables on both sides: $18 x=2 x-32$
We are tasked with solving the linear equation \(18x = 2x - 32\) for the variable \(x\). This is an equation where the variable \(x\) appears on both sides. To solve it, we need to isolate \(x\) on one side of the equation. We are given the following options to verify our result:
To begin, we gather all the terms containing the variable \(x\) on one side of the equation. We do this by subtracting \(2x\) from both sides of the equation:
\[18x - 2x = -32\]
Simplifying the expression on the left-hand side, we get:
\[16x = -32\]
Now that we have the equation in the form \(ax = b\), we isolate \(x\) by dividing both sides of the equation by the coefficient of \(x\), which is 16:
\[x = \frac{-32}{16}\]
Performing the division, we find:
\[x = -2\]
Comparing our result \(x = -2\) with the given options, we see that it matches option d. We can verify this by substituting \(x = -2\) back into the original equation:
Left-hand side: \(18 \times (-2) = -36\)
Right-hand side: \(2 \times (-2) - 32 = -4 - 32 = -36\)
Since both sides are equal, the value is correct. Therefore, the correct option is d, \(x = -2\). The final answer is \(\boxed{-2}\).
Linear equations with variables on both sides are useful in various real-world situations, such as comparing the costs of two different service providers. For instance, if Company A charges a flat fee of \(\$0\) and \(\$18\) per hour, while Company B charges a \(\$32\) flat fee but only \(\$2\) per hour, we can find the point at which their costs are equal by setting the equations equal: \(18h = 2h + 32\) (or \(18h - 2h = 32\)). Solving these simple algebraic equations allows you to make informed financial decisions in everyday life!