Question
In the following system of equation determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
$x - 2y = 8$
$5x - 10y = 10$
$5x - 10y = 10$
✓
Answer
Given
$x - 2y = 8$
$5x - 10y = 10$
To find: To determine whether the system has a uniqfue solution, no solution or infinitely many solutions
We know that the system of equations,
$a_1x + b_1y + c_1 = 0$
$a_2x + b_2y + c_2 = 0$
For unique solution
$\frac{\text{a}_1}{\text{a}_2}\neq\frac{\text{b}_1}{\text{b}_2}$
For no solution
$\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
For infinitely many solution
$\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$
Here,
$\frac{1}{5}=\frac{-2}{-10}=\frac{8}{10}$
$\frac{1}{5}=\frac{1}{5}\neq\frac{4}{5}$
Since, $\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$ which means $\frac{1}{5}=\frac{1}{5}\neq\frac{4}{5}$ hence the system of equation has no solution.
Hence the system of equation has no solution.
$x - 2y = 8$
$5x - 10y = 10$
To find: To determine whether the system has a uniqfue solution, no solution or infinitely many solutions
We know that the system of equations,
$a_1x + b_1y + c_1 = 0$
$a_2x + b_2y + c_2 = 0$
For unique solution
$\frac{\text{a}_1}{\text{a}_2}\neq\frac{\text{b}_1}{\text{b}_2}$
For no solution
$\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
For infinitely many solution
$\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$
Here,
$\frac{1}{5}=\frac{-2}{-10}=\frac{8}{10}$
$\frac{1}{5}=\frac{1}{5}\neq\frac{4}{5}$
Since, $\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$ which means $\frac{1}{5}=\frac{1}{5}\neq\frac{4}{5}$ hence the system of equation has no solution.
Hence the system of equation has no solution.
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