Solve the simultaneous equations:
$\begin{array}{l}
y=2 x-1 \text { and } y=x-1 . \\
x=\_, y=
\end{array}$
We are given a system of two linear equations in two variables, \(x\) and \(y\):
1) \(y = 2x - 1\)
2) \(y = x - 1\)
Our objective is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. Since both equations are already solved for \(y\), we can equate the expressions for \(y\) to find the value of \(x\).
To find the intersection point, we set the two expressions equal to each other:
\[2x - 1 = x - 1\]
To solve for \(x\), subtract \(x\) from both sides of the equation:
\[2x - x - 1 = -1\]
\[x - 1 = -1\]
Next, add 1 to both sides:
\[x = -1 + 1\]
\[x = 0\]
Now that we have found the value of \(x = 0\), we substitute this back into either of the original equations to find \(y\). Using the second equation, \(y = x - 1\):
\[y = 0 - 1\]
\[y = -1\]
Checking this result with the first equation, \(y = 2x - 1\):
\[y = 2(0) - 1 = 0 - 1 = -1\]
Both equations yield the same value for \(y\). Thus, the solution is \(x = 0\) and \(y = -1\).
This system of equations can represent the paths of two objects moving on a flat plane. For instance, imagine two people walking in a straight line: one follows a path represented by \(y = 2x - 1\) and the other follows the path \(y = x - 1\). Finding the point where these lines intersect, which we calculated as \((0, -1)\), tells us exactly where their paths meet. This type of calculation is fundamental in navigation, economics (finding market equilibrium), and engineering to determine intersection points in space.