Solve the equation.
$-9 x+1=-x+17$
The given equation is \(-9x + 1 = -x + 17\). To solve for \(x\), we need to isolate the variable \(x\) on one side of the equation and the constant terms on the other. This is a linear equation in one variable, which can be solved using standard algebraic manipulations.
First, we collect the terms involving \(x\) on one side. We can do this by adding \(9x\) to both sides of the equation:
\[-9x + 9x + 1 = -x + 9x + 17\]
\[1 = 8x + 17\]
Next, we isolate the constant terms by subtracting \(17\) from both sides:
\[1 - 17 = 8x\]
\[-16 = 8x\]
Now, we solve for \(x\) by dividing both sides by the coefficient of \(x\), which is \(8\):
\[x = \frac{-16}{8}\]
\[x = -2\]
Thus, the solution to the equation is \(x = -2\). Comparing this to the given choices, we see that it matches the second option.
Linear equations are fundamental in many real-life applications. For example, if a taxi company charges a flat fee of \(\$1\) plus \(\$9\) per mile, and another company charges a flat fee of \(\$17\) plus \(\$1\) per mile, setting up a linear equation like \(-9x + 1 = -x + 17\) (where \(x\) represents a distance or a variable factor) helps to find the "break-even" point where both services cost the same. This allows individuals to make informed financial decisions based on their specific needs.