The linear equation `2x-5y=7` has

Options

A. a unique solution
B. two solutions
C. infinitely many solutions
D. no solution

Explanation

To determine the nature of the solutions for the linear equation \(2x - 5y = 7\), we can analyze it step by step.

### Step 1: Identify the equation
The given linear equation is:
\[ 2x - 5y = 7 \]

### Step 2: Understand the concept of solutions
A linear equation in two variables (like this one) represents a straight line on a coordinate plane. Any point \((x, y)\) that lies on this line is a solution to the equation. Since a line extends infinitely in both directions, there are infinitely many points (solutions) that satisfy the equation.

### Step 3: Find specific solutions
To illustrate that there are infinitely many solutions, we can substitute different values for \(x\) and solve for \(y\).

1. **Let \(x = 1\)**:
\[ 2(1) - 5y = 7 \implies 2 - 5y = 7 \implies -5y = 7 - 2 \implies -5y = 5 \implies y = -1 \]
So, one solution is \((1, -1)\).

2. **Let \(x = 0\)**:
\[ 2(0) - 5y = 7 \implies -5y = 7 \implies y = -\frac{7}{5} \]
So, another solution is \((0, -\frac{7}{5})\).

### Step 4: Conclusion
Since we can keep substituting different values for \(x\) and find corresponding values for \(y\), we conclude that the linear equation \(2x - 5y = 7\) has infinitely many solutions.

### Final Answer
The linear equation \(2x - 5y = 7\) has **infinitely many solutions**.

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