Write a pair of linear equations whose unique solution is \(x = -1\), \(y = 3\). How many such pairs can you write?
Infinitely many. One example: \(x + y = 2\) and \(2x - y = -5\).
We are told that the solution is \(x = -1\) and \(y = 3\).
Step 1: Write the general form of a linear equation in two variables:
\(a x + b y = c\)
Step 2: Substitute the values \(x = -1\) and \(y = 3\).
\(a(-1) + b(3) = c\)
\(-a + 3b = c\)
So any equation of the form \(a x + b y = -a + 3b\) will be satisfied by the point \((-1,3)\).
Step 3: To form a pair of equations, we need two different equations. Let us take:
Equation 1: \(x + y = 2\)
Equation 2: \(2x - y = -5\)
Step 4: Check if \(x = -1, y = 3\) satisfies both.
For Equation 1: \((-1) + 3 = 2\) ✔
For Equation 2: \(2(-1) - 3 = -2 - 3 = -5\) ✔
Since both are true, the pair \(x + y = 2\) and \(2x - y = -5\) has the solution \((-1,3)\).
Step 5: How many such pairs can we write?
There are infinitely many choices because we can pick any two different equations of the form \(a x + b y = -a + 3b\), as long as the two equations are not multiples of each other (so that the lines are not parallel).