Question
Show that the paths represented by the equations $x-3 y=2$ and $-2 x+6 y=5$ are parallel.
✓
Answer
The given system of equations can be written as follows:
$x-3 y-2=0 \text { and }-2 x+6 y-5=0$
The given equations are of the following form:
$a_1 x+b_1 y+c_1=0 \text { and } a_2 x+b_2 y+c_2=0$
Here, $a_1=1, b_1=-3, c_1=-2$ and $a_2=-2, b_2=6$ and $c_2=-5$
$\therefore\frac{\text{a}_1}{\text{a}_2}=\frac{1}{-2}=\frac{-1}{2},\ \frac{\text{b}_1}{\text{b}_2}=\frac{-3}{6}=\frac{-1}{2}$ and $\frac{\text{c}_1}{\text{c}_2}=\frac{-2}{-5}=\frac{2}{5}$
For inconsistency, we must have:
$\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
Hence, the paths represented by the equations are parallel.
$x-3 y-2=0 \text { and }-2 x+6 y-5=0$
The given equations are of the following form:
$a_1 x+b_1 y+c_1=0 \text { and } a_2 x+b_2 y+c_2=0$
Here, $a_1=1, b_1=-3, c_1=-2$ and $a_2=-2, b_2=6$ and $c_2=-5$
$\therefore\frac{\text{a}_1}{\text{a}_2}=\frac{1}{-2}=\frac{-1}{2},\ \frac{\text{b}_1}{\text{b}_2}=\frac{-3}{6}=\frac{-1}{2}$ and $\frac{\text{c}_1}{\text{c}_2}=\frac{-2}{-5}=\frac{2}{5}$
For inconsistency, we must have:
$\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
Hence, the paths represented by the equations are parallel.
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