Answered

What is the solution to the system of equations?

$\begin{array}{l}

2 x-y=7 \\

y=2 x+3

\end{array}$


A$(2,3)$
B$(2,7)$
Cno solution
Dinfinite number of solutions

Verified Answer :

  • Analyze the system: \(2x - y = 7\) and \(y = 2x + 3\).
  • Substitute the second equation into the first: \(2x - (2x + 3) = 7\).
  • Simplify to find a contradiction: \(-3 = 7\).
  • Conclude that there is \(\boxed{\text{no solution}}\).

Explanation

  1. Analysis of the System of Equations

We are presented with a system of two linear equations:
1) \(2x - y = 7\)
2) \(y = 2x + 3\)
Our goal is to determine the solution to this system. A system of two linear equations can either have a unique solution (intersecting lines), no solution (parallel distinct lines), or an infinite number of solutions (coincident lines). We will analyze this system to determine which case applies here.

  1. Applying the Substitution Method

To solve this system, we can use the substitution method since the second equation is already explicitly solved for \(y\). We substitute the expression \(y = 2x + 3\) into the first equation:
\(2x - (2x + 3) = 7\)
Expanding this equation gives:
\(2x - 2x - 3 = 7\)
Simplifying the left side:
\(-3 = 7\)
This result is a contradiction, as \(-3\) cannot equal \(7\).

  1. Interpretation of the Result and Final Conclusion

The resulting equation \(-3 = 7\) is always false, regardless of the value of \(x\). This indicates that there is no value of \(x\) (and consequently no value of \(y\)) that can satisfy both equations simultaneously. Geometrically, this means that the two lines represented by the equations are parallel and do not intersect. Therefore, the system has no solution.

Examples

In real-world applications, systems of linear equations often represent conflicting constraints. For example, if you are planning two production schedules for a factory where one line demands that the total daily cost be at a certain level based on output, but another line represents a budget constraint that is mathematically incompatible with the first, the factory management will find that there is 'no solution' to meet both conditions simultaneously. This mathematical identification of conflict prevents the pursuit of impossible production targets.

Related Questions