Solve the system of equations:
$\begin{array}{l}-1=2 x-y \\ 8 x-4 y=-4\end{array}$
The problem requires us to analyze the given system of two linear equations in variables \(x\) and \(y\):
We need to determine the solution to this system. A system of two linear equations can have either a unique solution, no solution, or infinitely many solutions. We can investigate this by comparing the equations directly.
Let's look at the second equation: \(8x - 4y = -4\).
Notice that all coefficients and the constant on the right-hand side are divisible by 4.
Dividing the entire equation \(8x - 4y = -4\) by \(4\), we get:
\(\frac{8x}{4} - \frac{4y}{4} = \frac{-4}{4}\)
\(2x - y = -1\)
This resulting equation is identical to the first equation in our system. Therefore, the two equations are not distinct; they represent the same line in the coordinate plane.
Since both equations describe the same line, any pair \((x, y)\) that satisfies the first equation will also satisfy the second. This means the system has infinitely many solutions.
To describe these solutions, we can express \(y\) in terms of \(x\). Starting from \(2x - y = -1\), we add \(y\) to both sides and add \(1\) to both sides, which gives:
\(y = 2x + 1\)
Thus, the set of all solutions can be written as the set of ordered pairs \((x, 2x + 1)\) for any real number \(x\). Alternatively, we can use a parameter \(t\) such that \(x = t\) and \(y = 2t + 1\).
In real life, systems of equations like this appear when two different ways of calculating a cost or a requirement yield the exact same result. For example, if two different subscription plans for a service, after adjusting for their specific constraints, turn out to be identical, any amount of usage would lead to the same cost, meaning there isn't a single "better" plan but rather a continuous range of equivalent options. This happens in engineering or resource management when equations derived from different physical constraints simplify to the same relationship.