Verified Answer :

  • Set the two equations equal to each other to solve for \(x\): \(2x - 1 = -x + 5\).
  • Simplify and solve for \(x\): \(3x = 6 \implies x = 2\).
  • Substitute \(x = 2\) back into one of the equations to find \(y\): \(y = 2(2) - 1 = 3\).
  • Final Answer: \(x = 2, y = 3\). \(\boxed{x=2, y=3}\)

Explanation

  1. Problem Analysis

The problem asks us to solve the system of two linear equations:
\[y = 2x - 1\]
\[y = -x + 5\]
We are looking for the pair \((x, y)\) that satisfies both equations simultaneously. Since both expressions are set equal to \(y,\) we can equate them to find the value of \(x.\)

  1. Solving for x

We equate the two expressions for \(y:\)
\[2x - 1 = -x + 5\]
To solve for \(x,\) we move all terms involving \(x\) to one side and the constant terms to the other side:
\[2x + x = 5 + 1\]
\[3x = 6\]
Dividing both sides by 3, we find the value of \(x:\)
\[x = \frac{6}{3} = 2\]

  1. Solving for y and Verification

Now that we have the value of \(x = 2,\) we substitute it into one of the original equations to find the corresponding value of \(y.\) Using the first equation:
\[y = 2(2) - 1\]
\[y = 4 - 1\]
\[y = 3\]
We can verify this with the second equation to ensure accuracy:
\[y = -(2) + 5\]
\[y = 3\]
Since both equations yield the same value for \(y,\) our solution is consistent. Thus, \(x = 2\) and \(y = 3.\)

Examples

In real life, this system of equations can represent the paths of two objects. For instance, imagine two people walking on paths defined by these functions on a grid: one person follows the path \(y = 2x - 1\) and the other follows the path \(y = -x + 5\). Solving the system allows us to find the exact point \((2, 3)\) where their paths intersect. This principle is vital in fields like navigation, traffic control, and urban planning where it is necessary to determine if or where two trajectories might cross.

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