Pair of Linear Equations with a Given Unique Solution

Video Explanation

Question

Write a pair of linear equations which has the unique solution \[ x = -1,\quad y = 3. \]

How many such pairs can you write?

Solution

Step 1: Use the Given Solution

Since the solution is \[ (x,\; y) = (-1,\; 3), \] the required equations must be satisfied by these values.

Step 2: Write One Pair of Linear Equations

Choose any two linear equations that are true for \(x = -1\) and \(y = 3\).

For example:

\[ x + y = 2 \]

\[ 2x + y = 1 \]

Verification:

\[ -1 + 3 = 2 \quad \checkmark \]

\[ 2(-1) + 3 = 1 \quad \checkmark \]

Hence, the pair

\[ x + y = 2,\quad 2x + y = 1 \]

has the unique solution \(x = -1,\; y = 3\).

Step 3: Number of Such Pairs

We can form infinitely many linear equations passing through the point \((-1,\;3)\).

By choosing any two such non-parallel equations, we get a pair having the same unique solution.

Conclusion

One such pair of linear equations is:

\[ x + y = 2,\quad 2x + y = 1 \]

The number of such pairs is:

\[ \boxed{\text{Infinitely many}} \]

\[ \therefore \quad \text{Infinitely many pairs of linear equations can be written.} \]