Question

The equations $2x+y=5$ and $x+2y=4$ are:
A. Consistent and have a unique solution.
B. Consistent and have infinitely many solutions.
C. Inconsistent.
D. None of these.

Answer 1

Hint: The consistency of a pair of equations ${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$ and ${{a}_{2}}x+{{b}_{2}}y={{c}_{2}}$ , is determined as follows:
 $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ : The system is consistent and has a unique solution.
The pair of equations represent a pair of intersecting lines.
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}$ : The system is consistent and has infinitely many solutions.
The pair of equations represent the same line.
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$ : The system is consistent, i.e. it has no solutions.
The pair of equations represent a pair of parallel lines.

Complete step-by-step answer:
Comparing the pair of equations $2x+y=5$ and $x+2y=4$ with equations ${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$ and ${{a}_{2}}x+{{b}_{2}}y={{c}_{2}}$ , we can say that ${{a}_{1}}=2,{{b}_{1}}=1,{{c}_{1}}=5$ and ${{a}_{2}}=1,{{b}_{2}}=2,{{c}_{2}}=4$ .
∴ $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{2}{1},\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{1}{2},\dfrac{{{c}_{1}}}{{{c}_{2}}}=\dfrac{5}{4}$ .
Since, $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ , the pair of equations is consistent and has a unique solution.
The correct answer option is A. Consistent and have a unique solution.

Note: The solutions to the equations ${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$ and ${{a}_{2}}x+{{b}_{2}}y={{c}_{2}}$ can be determined using Cramer's rule:
 $D=\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} \\ {{b}_{1}} & {{b}_{2}} \\ \end{matrix} \right|$ , ${{D}_{x}}=\left| \begin{matrix} {{c}_{1}} & {{c}_{2}} \\ {{b}_{1}} & {{b}_{2}} \\ \end{matrix} \right|$ , ${{D}_{y}}=\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} \\ {{c}_{1}} & {{c}_{2}} \\ \end{matrix} \right|$ .
The values of the variables x and y are given by: $x=\dfrac{{{D}_{x}}}{D}$ and $y=\dfrac{{{D}_{y}}}{D}$ .
If, D = 0, the system is:
Either consistent and has infinitely many solutions. In this case ${{D}_{x}}={{D}_{y}}=0$ .
Or it is inconsistent, i.e. it has no solutions. In this case ${{D}_{x}} \ne 0$ and ${{D}_{y}} \ne 0$ .
Consistent equations with infinitely many solutions are also known as dependent equations, because both are multiples of the same equation.