Question

Write the equation of the line that is parallel to $y = x + 3$ and passes through $\left( { - 4,1} \right)$?

Answer 1

Hint: The above question is a simple question of linear equations in two variables. The general equation of the slope-intercept form of the line is given as \[y = mx + c\], where m is the slope of the line and c is the y-intercept of the line. Also, note that when a line is parallel to the x-axis then its slope is equal to 0, so the equation of such line is given as \[y = a\] where a is the y-intercept that line.

Complete step by step answer:
We can see from the question that we are provided with a line that is parallel to the x-axis.
The slope of the line $y = x + 3$ is,
$ \Rightarrow m = 1$
Since we know that when a line is parallel to the line $y = x + 3$ then its slope is equal to 1.
Also, we know that the slope-intercept form of the line is given by \[y = mx + c\], where m is the slope of the line and c is the y-intercept of the line.
So, we can say that equation of the line is equal to,
$ \Rightarrow y = 1 \times x + c$
Simplify the terms,
$ \Rightarrow y = x + c$
Now, we know from the question that the line \[y = x + c\] passes through the point (-4, 1).
So, the point (-4, 1) will satisfy the line \[y = x + c\]. Substitute the value in \[y = x + c\].
$ \Rightarrow 1 = - 4 + c$
Move the constant part on the other side,
$ \Rightarrow c = 5$
Substitute the values in the above equation,
$\therefore y = x + 5$
Hence, $y = x + 5$ is the equation of the line passing through the point (-4, 1) which is parallel to the line $y = x + 3$.

Note: The general equation of the line is given as $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ where m is the slope of the line and $\left( {{y_1},{x_1}} \right)$ is the point through which the line passes. We know that the line given in the question is parallel to the line $y = x + 3$, so the slope of the line is equal to 1 and since the line passes through the point (-4, 1), hence the equation of line will be:
$ \Rightarrow y - 1 = 1\left( {x - \left( { - 4} \right)} \right)$
Simplify the terms,
$ \Rightarrow y - 1 = x + 4$
Move the constant part on the other side and add,
$\therefore y = x + 5$
Hence, $y = x + 5$ is the equation of the line.