Mathematics

Problem

Study the following pair of equations containing the parameter

m

\begin{aligned}(m+1)x+2y&=8,\\3x+(m+1)y&=10.\end{aligned}

Let

R(2,2)

be a point in the plane. a. (i) Derive a condition on

m

so that

R

satisfies the first equation. b. (ii) Derive a condition on

m

so that

R

satisfies the second equation. c. (iii) Hence find the value(s) of

m

for which

R

satisfies both equations simultaneously. d. (iv) For the value of

m

obtained in (iii), compare the ratios

\dfrac{a_1}{a_2}

and

\dfrac{b_1}{b_2}

of the coefficients in the two equations, and state whether the system has a unique solution, no solution, or infinitely many solutions.