Write a pair of linear equations which has the unique solution `x= -1, y=3`. How many such pairs can you write?

Answer

To find a pair of linear equations that has a unique solution at \( x = -1 \) and \( y = 3 \), we can follow these steps:

### Step 1: Use the point to create equations
We know that the solution \( (x, y) = (-1, 3) \) satisfies both equations. We can start by forming two linear equations in the standard form \( ax + by = c \).

### Step 2: Create the first equation
Let’s choose coefficients for \( x \) and \( y \). For example, we can take:
- \( a_1 = 3 \) (coefficient of \( x \))
- \( b_1 = 5 \) (coefficient of \( y \))

Using the point \( (-1, 3) \):
\[ 3(-1) + 5(3) = c_1 \]
Calculating this gives:
\[ -3 + 15 = 12 \quad \Rightarrow \quad c_1 = 12 \]
Thus, the first equation is:
\[ 3x + 5y = 12 \quad \text{(Equation 1)} \]

### Step 3: Create the second equation
Now, let’s choose different coefficients for the second equation. For example, we can take:
- \( a_2 = 2 \) (coefficient of \( x \))
- \( b_2 = 3 \) (coefficient of \( y \))

Using the point \( (-1, 3) \):
\[ 2(-1) + 3(3) = c_2 \]
Calculating this gives:
\[ -2 + 9 = 7 \quad \Rightarrow \quad c_2 = 7 \]
Thus, the second equation is:
\[ 2x + 3y = 7 \quad \text{(Equation 2)} \]

### Step 4: Verify the uniqueness of the solution
To ensure that the equations have a unique solution, we check the condition:
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
Calculating this:
\[ \frac{3}{2} \neq \frac{5}{3} \]
Since the ratios are not equal, the equations have a unique solution.

### Step 5: Conclusion
The pair of linear equations we found is:
1. \( 3x + 5y = 12 \)
2. \( 2x + 3y = 7 \)

### How many such pairs can you write?
We can create infinitely many pairs of linear equations that have the same unique solution by multiplying the equations by any non-zero constant. For example, if we multiply the first equation by 2 and the second by 3, we would still have equations that intersect at the same point \( (-1, 3) \).