Solve the simultaneous equations 3x - y = 5, 4x - 3y = -1. Hence, find p, if y = px - 3.

Given,

3x - y = 5 ……(i)

4x - 3y = -1 …..(ii)

Solving (i) we get,

⟹ y = 3x - 5 …..(iii)

Substituting value of y from eqn. (iii) in eqn. (ii) we get,

⟹ 4x - 3(3x - 5) = -1

⟹ 4x - 9x + 15 = -1

⟹ -5x = -1 - 15

⟹ -5x = -16

⟹ x = $\dfrac{16}{5}$.

Substituting value of x in eqn. (iii) we get,

$\Rightarrow y = 3 \times \dfrac{16}{5} - 5 \\[1em] = \dfrac{48}{5} - 5 \\[1em] = \dfrac{48 - 25}{5} \\[1em] = \dfrac{23}{5}$

Given, y = px - 3. Substituting value of x and y in equation,

$\Rightarrow \dfrac{23}{5} = \dfrac{16}{5}p - 3 \\[1em] \Rightarrow \dfrac{23}{5} = \dfrac{16p - 15}{5} \\[1em] \Rightarrow 23 = 16p - 15 \\[1em] \Rightarrow 16p = 38 \\[1em] \Rightarrow p = \dfrac{38}{16} = \dfrac{19}{8}.$

Hence, x =$\dfrac{16}{5},\text{ y }= \dfrac{23}{5}\text{ and p} = \dfrac{19}{8}.$