Question
Show that the paths represented by the equations $x - 3y = 2$ and $-2x + 6y = 5$ are parallel.
✓
Answer
The given system of equations can be written as follows:
$x - 3y - 2 = 0$ and $-2x + 6y - 5 = 0$
The given equations are of the following form:
$a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$
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.
$x - 3y - 2 = 0$ and $-2x + 6y - 5 = 0$
The given equations are of the following form:
$a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$
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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