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:
$2x + y = 5$
$4x + 2y = 10$
✓
Answer
The given equations are,
$2x + y = 5 .....(i)$
$4x + 2y = 10 .......(ii)$
The given equations are of the from,
$a_1x + b_1y + c_1 = 0$
$a_2x + b_2y + c_2 = 0$
Where $a_1 = 2, b_1 = 1, c_1= -5$
$a_2 = 4, b_2 = 2$ and $c_2 = -10$
We have,
$\frac{\text{a}_1}{\text{a}_2}=\frac{2}{4}=\frac{1}{2},$
$\frac{\text{b}_1}{\text{b}_2}=\frac{1}{2},$
$\frac{\text{c}_1}{\text{c}_2}=\frac{-5}{-10}=\frac{1}{2}$
$\therefore\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$
Thus, the given system of equations has infinitely many solution.
$2x + y = 5 .....(i)$
$4x + 2y = 10 .......(ii)$
The given equations are of the from,
$a_1x + b_1y + c_1 = 0$
$a_2x + b_2y + c_2 = 0$
Where $a_1 = 2, b_1 = 1, c_1= -5$
$a_2 = 4, b_2 = 2$ and $c_2 = -10$
We have,
$\frac{\text{a}_1}{\text{a}_2}=\frac{2}{4}=\frac{1}{2},$
$\frac{\text{b}_1}{\text{b}_2}=\frac{1}{2},$
$\frac{\text{c}_1}{\text{c}_2}=\frac{-5}{-10}=\frac{1}{2}$
$\therefore\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$
Thus, the given system of equations has infinitely many solution.
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