Question

(a) Check whether the line $3x-2y+9=0$ pass through the point $\left( 1,6 \right)$.
(b) Write down the equation of the line through $\left( 3,7 \right)$ and of slope $\dfrac{3}{2}$.
(c) Show that the lines mentioned above are parallel.

Answer 1

Hint:
(a)Put the values of coordinates in the equation of line and if it satisfies the equation then the line passes through the given point otherwise not.
(b) Equation of a line passing through a point $\left( a,b \right)$ and having slope $m$ is given by as: $y-b=m\left( x-a \right)$.
(c) To show that the given lines are parallel, show that the slopes of the two lines are the same.

Complete step-by-step answer:

An equation of a line in the $x-y$ plane represents the relation between variables, $x\text{ and }y$. It represents the path followed by the line in that plane. Now, in a plane a line passes through infinite points. If a line passes through a particular point then the given point must satisfy the equation of line.
If we have to form an equation of a line passing through a point $\left( a,b \right)$ and having slope $m$, we use the formula: $y-b=m\left( x-a \right)$.
Parallel lines are lines having the same slope. To show that the lines are parallel to each other, we just have to show that they have the same slope.
Now, let us come to the question.
(a) Substituting the given coordinate in the line we get, $3\times 1-2\times 6+9=3-12+9=0$. Therefore, the point satisfies the equation of line and hence the line passes through the point.
(b) Equation of the line passing through $\left( 3,7 \right)$ and having slope $\dfrac{3}{2}$ is,
$\begin{aligned} & y-7=\dfrac{3}{2}\left( x-3 \right) \\\ & 2y-14=3x-9 \\\ & 2y-3x-5=0 \\\ \end{aligned}$
(c) Now, slope of the line $3x-2y+9=0$ is given as: $slope=\dfrac{\text{-coefficient of x}}{\text{coefficient of y}}=\dfrac{-3}{-2}=\dfrac{3}{2}$. We can see that the slope of the two lines are equal. Therefore, the two lines are parallel.

Note: A line satisfies all the points that comes in its path, therefore it is the only method we can apply to check whether a given point lies on the line or not. The slope of a line denotes the angle it forms with the positive direction of $x-axis$ and the slope of parallel lines are always the same. That means, parallel lines form the same angle with positive direction of $x-axis$.