How many linear equations in x and y can be satisfied by `x=1` and `y=2` ?

Options

A. Only one
B. Two
C. Infinitely many
D. Three

Explanation

To determine how many linear equations in two variables \(x\) and \(y\) can be satisfied by the point \( (1, 2) \), we can follow these steps:

### Step 1: Understand the Concept of Linear Equations
A linear equation in two variables can be expressed in the form:
\[ ax + by + c = 0 \]
where \(a\), \(b\), and \(c\) are constants. The solution to this equation is a point \((x, y)\) that lies on the line represented by the equation.

### Step 2: Substitute the Given Values
We are given the values \(x = 1\) and \(y = 2\). We can substitute these values into the general form of the linear equation:
\[ a(1) + b(2) + c = 0 \]
This simplifies to:
\[ a + 2b + c = 0 \]

### Step 3: Analyze the Equation
The equation \(a + 2b + c = 0\) represents a linear relationship among the constants \(a\), \(b\), and \(c\). Since there are three variables (a, b, and c) but only one equation, we have the freedom to choose values for two of the variables, while the third will be determined by the equation.

### Step 4: Determine the Number of Solutions
Since we can choose any values for two of the variables \(a\), \(b\), or \(c\), there are infinitely many combinations that will satisfy the equation. This means that there are infinitely many linear equations that can pass through the point \((1, 2)\).

### Conclusion
Thus, the answer to the question is that there are infinitely many linear equations in \(x\) and \(y\) that can be satisfied by the point \((1, 2)\).

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