Question

How do you find the equation of the line parallel to $y = - 5x + 3$ and passing through $\left( {4,3} \right)$ ?

Answer 1

Here in this question, we have to find the equation of the line parallel to the given equation, remember when lines are parallel the slope of both the equation is same and $\left( {{x_1},{y_1}} \right)$ is the point where the line passes. This can be solved by using slope-intercept form i.e., $y = mx + b$. On simplification we get the required solution.

Complete step by step solution:
Remember, parallel lines are lines that never intersect. Because of this, a pair of parallel lines have to have the same slope, but different intercepts (if they had the same intercepts, they would be identical lines).So, to find an equation of a line that is parallel to another, we have to make sure both equations have the same slope. In the general equation of a line $y = mx + b\;$, the $m$ represents your slope value.

Given the equation of the parallel line $y = - 5x + 3$ has a slope of $m = - 5$. We can find the equation of line using the slope-intercept form. The slope-intercept form of a linear equation is: $y = mx + b$-------(1)
Where $m$ is the slope and $b$ is the y-intercept value.
Parallel lines always have the same slope hence the slope of the line is also -5. Substitute the slope $m$ and the values of the point from the problem for $x = 4$ and $y = 3$ and solve for $b$:
Equation (1) becomes
$3 = \left( { - 5} \right)4 + b$
$\Rightarrow \,\,\,3 = - 20 + b$
Solve for b
$b = 3 + 20$
$\Rightarrow \,\,\,b = 23$
We can substitute for $m$ and $b$ in the equation (1) to find the equation of the line:
$\therefore\,\,\,y = - 5x + 23$

Hence, the equation of the line is $y = - 5x + 23$.

Note: To determine the equation of line we use the slope-intercept formula. While multiplying the terms we must take care of signs, and we should know about the sign conventions. On further simplification we must know about the simple arithmetic operations and the table of multiplication is needed.