Write a pair of linear equations which has the unique solution `x=-1,y=3`

Answer

To find a pair of linear equations that has a unique solution at the point \( x = -1 \) and \( y = 3 \), 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:
\[ Ax + By = C \]
where \( A \), \( B \), and \( C \) are constants.

### Step 2: Substitute the known solution into the equation
Since we know that \( x = -1 \) and \( y = 3 \) is a solution, we can substitute these values into the equation to find \( C \).

### Step 3: Choose values for \( A \) and \( B \)
Let's choose arbitrary values for \( A \) and \( B \). For example, let’s take:
- \( A = 1 \)
- \( B = 4 \)

### Step 4: Calculate \( C \)
Substituting \( x = -1 \) and \( y = 3 \) into the equation:
\[ 1(-1) + 4(3) = C \]
Calculating this gives:
\[ -1 + 12 = C \]
\[ C = 11 \]
So, one equation is:
\[ x + 4y = 11 \]

### Step 5: Create a second equation
Now, we need another equation that also intersects at the point \( (-1, 3) \). Let’s choose different values for \( A \) and \( B \). For example:
- \( A = 3 \)
- \( B = 2 \)

### Step 6: Calculate \( C \) for the second equation
Substituting \( x = -1 \) and \( y = 3 \) into the second equation:
\[ 3(-1) + 2(3) = C \]
Calculating this gives:
\[ -3 + 6 = C \]
\[ C = 3 \]
So, the second equation is:
\[ 3x + 2y = 3 \]

### Final Pair of Linear Equations
Thus, the pair of linear equations that has a unique solution at \( (-1, 3) \) is:
1. \( x + 4y = 11 \)
2. \( 3x + 2y = 3 \)