Question
For what value of $$p,$$ is the system of equations :
$$\eqalign{
& {p^3}x + {\left( {p + 1} \right)^3}y = {\left( {p + 2} \right)^3} \cr
& px + \left( {p + 1} \right)y = p + 2 \cr
& x + y = 1 \cr} $$
Consistent ?
A.
$$p = 0$$
B.
$$p = 1$$
C.
$$p = - 1$$
D.
For all $$p > 1$$
Answer :
$$p = - 1$$
Solution :
The given system of equations are :
$$\eqalign{
& {p^3}x + {\left( {p + 1} \right)^3}y = {\left( {p + 2} \right)^3}\,\,\,\,.....\left( 1 \right) \cr
& px + \left( {p + 1} \right)y = \left( {p + 2} \right)\,\,\,\,\,\,.....\left( 2 \right) \cr
& x + y = 1\,\,\,\,\,\,\,.....\left( 3 \right) \cr} $$
This system is consistent, if values of $$x$$ and $$y$$ from first two equation satisfy the third equation.
which \[ \Rightarrow \,\left| {\begin{array}{*{20}{c}}
{{p^3}}&{{{\left( {p + 1} \right)}^3}}&{{{\left( {p + 2} \right)}^3}}\\
p&{\left( {p + 1} \right)}&{\left( {p + 2} \right)}\\
1&1&1
\end{array}} \right| = 0\]
\[ \Rightarrow \,\left| {\begin{array}{*{20}{c}}
{{p^3}}&{{{\left( {p + 1} \right)}^3} - {p^3}}&{{{\left( {p + 2} \right)}^3} - {p^3}}\\
p&1&2\\
1&0&0
\end{array}} \right| = 0\]
$$\eqalign{
& \Rightarrow \,2{\left( {p + 1} \right)^3} - 2{p^3} - {\left( {p + 2} \right)^3} + {p^3} = 0 \cr
& \Rightarrow \,2\left( {{p^3} + 1 + 3{p^2} + 3p} \right) - 2{p^3} - \left( {{p^3} + 8 + 12p + 6{p^2}} \right) + {p^3} = 0 \cr
& \Rightarrow \,2{p^3} + 2 + 6{p^2} + 6p - 2{p^3} - {p^3} - 8 - 12p - 6{p^2} + {p^3} = 0 \cr
& \Rightarrow \, - 6 - 6p = 0 \cr
& \Rightarrow \,p = - 1 \cr} $$