What are the solution(s) of $-\frac{1}{2} x+4=x+1$ ?
The given equation is \(-\frac{1}{2}x + 4 = x + 1\). This is a linear equation in one variable, \(x\). Our goal is to isolate \(x\) to determine its value. A linear equation of the form \(ax + b = cx + d\) can be solved by grouping all terms containing \(x\) on one side and all constant terms on the other side.
To isolate \(x\), we first collect all terms with \(x\) on one side of the equation. We can add \(\frac{1}{2}x\) to both sides of the equation:
\[-\frac{1}{2}x + \frac{1}{2}x + 4 = x + \frac{1}{2}x + 1\]
\[4 = \frac{3}{2}x + 1\]
Next, we move the constant term \(1\) to the left side by subtracting \(1\) from both sides:
\[4 - 1 = \frac{3}{2}x\]
\[3 = \frac{3}{2}x\]
Now, we solve for \(x\) by multiplying both sides by the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\):
\[3 \times \frac{2}{3} = \left(\frac{3}{2}x\right) \times \frac{2}{3}\]
\[2 = x\]
We can verify this result by substituting \(x = 2\) back into the original equation:
\[-\frac{1}{2}(2) + 4 = -1 + 4 = 3\]
\[2 + 1 = 3\]
Since both sides are equal to \(3\), our solution \(x = 2\) is correct. Comparing this to the given choices, the value matches the third option.
Linear equations are fundamental in many real-life applications. For example, consider a taxi service that charges a flat fee plus a rate per mile. If Company A charges \(\$4\) base plus \(\$0.50\) per mile, and Company B charges \(\$1\) base plus \(\$1.00\) per mile, you could set up an equation \(4 - 0.5x = 1 + x\) to determine after how many miles both companies cost exactly the same. Solving this equation helps you make informed financial decisions when comparing service providers.