The expression ax + by has value 7 when x = 2 and y = 1. It has value 1 when x = -1 and y = 1. Find a and b.

When x = 2 and y = 1,

⇒ ax + by = 7

⇒ a $\times$ 2 + b $\times$ 1 = 7

⇒ 2a + b = 7 ……………….(1)

When x = -1 and y = 1,

⇒ a $\times$ (-1) + b $\times$ 1 = 1

⇒ -a + b = 1 ……………….(2)

Subtracting equation (2) from equation (1), we get:

$\begin{matrix} & 2a & + & b & = & 7 \ & -a & + & b & = & 1 \ & + & - & & & - \ \hline & 3a & & & = & 6 \ \Rightarrow & a & & & = & \dfrac{6}{3} \ \end{matrix}$

⇒ a = 2

Substituting the value of a in equation (2), we get:

⇒ -2 + b = 1

⇒ b = 1 + 2 = 3

Hence, a = 2 and b = 3.