Question

Equation of the line passing through $(1,2)$ and parallel to the line $y=3x-1$ is
1. $y+2=x+1$
2. $y+2=3(x+1)$
3. $y-2=3(x-1)$
4. $y-2=x-1$

Answer 1

To solve this problem, first we will use the slope intercept form of the straight line to find the slope of the given equation and then we will find the slope of the required line by using parallel lines properties and use the point slope form to find the equation of the required line that is parallel to the given line.

Complete step-by-step solution:
If the distance between two lines is always the same, the lines are said as parallel lines. Parallel lines never meet each other (i.e. they never intersect or cross each other’s path). Slopes of parallel lines are always equal.
Now, we will understand the slope intercept form and point slope form:
Slope-intercept form is the general form of the straight line equation. It is represented as: $y=mx+c$ where $c$ is the intercept and $m$ is the slope, that’s why it is called slope intercept form. The value of $m$ and $c$ are real numbers. The slope of the line is also termed as gradient.
Point slope form is one of the more commonly used forms of a linear equation, and has the following structure: $y-{{y}_{1}}=m(x-{{x}_{1}})$ where $m$ is the slope of the line and $({{x}_{1}},{{y}_{1}})$ is a point on the line. Point slope form is used when one point of the line and the slope are known.
Now, according to the question:
The equation of the given line is: $y=3x-1$
As when we compare the given equation with the slope-intercept form (i.e. $y=mx+c$ ) , we will get slope $m=3$
Also, it is given that the required line is parallel to the given line.
So, slope of the required line will also be $m=3$
It is also given that the required line passes through the point $(1,2)$
Thus, we consider that:
${{x}_{1}}=1$
${{y}_{1}}=2$
Now, as per the point slope form of a straight line, the equation of required line with slope $m=3$ and passing through the given point $({{x}_{1}}=1,{{y}_{1}}=2)$ , will be given by:
$y-{{y}_{1}}=m(x-{{x}_{1}})$
$\Rightarrow y-2=3(x-1)$
So, the required equation of line is: $y-2=3(x-1)$
Hence, the correct option is $3$.

Note: A single existing line can have an infinite number of parallel lines. We come across parallel lines in daily life, such as we see parallel lines in railway tracks, zebra crossing, stairs, opposite walls, window linings, lines in notebook, opposite sides of blackboard and whiteboard.