Question

How do you find the equation of the line that passes through point $\left( {4, - 1} \right)$ and has a slope of $m = - 5$ ?

Answer 1

Here in this question, we have to find the equation of the line given. This can be solve by using the point-slope formula i.e., $y - {y_1} = m\left( {x - {x_1}} \right)$ where m is the slope and $\left( {{x_1},{y_1}} \right)$ is the point where the line passes. And can also solve this by using slope-intercept form i.e., $y = mx + b$. On simplification we get the required solution.

Complete step by step solution:
Given the line which passes through the point $\left( {4, - 1} \right)$ and the line has a slope of $m = - 5$.
Method 1:
Now, we have to find the equation of the line which passes through the point $\left( {4, - 1} \right)$ and the slope $m = - 5$ by using the slope-point formula $y - {y_1} = m\left( {x - {x_1}} \right)$-------(1)
Substitute the slope m and the point $\left( {{x_1},{y_1}} \right) = \left( {4, - 1} \right)$ in the point slope formula.Consider the equation (1)
$y - {y_1} = m\left( {x - {x_1}} \right)$
Where $m = - 5$, ${x_1} = 4$ and ${y_1} = - 1$ on substitution, we get
$\Rightarrow \,\,y - \left( { - 1} \right) = - 5\left( {x - 4} \right)$
$\Rightarrow \,\,y + 1 = - 5x + 20$
Subtract 1 on both side, then
$y + 1 - 1 = - 5x + 20 - 1$
On simplification, we get
$\therefore\,\,y = - 5x + 19$

Method 2:
We can also solve this for the slope-intercept form. The slope-intercept form of a linear equation is: $y = mx + b$-------(2)
Where $m$ is the slope and $b$ is the y-intercept value.
Substitute the slope from the problem for $m$ and the values of the point from the problem for $x = 4$ and $y = - 1$ and solve for $b$:
Equation (2) becomes
$- 1 = \left( { - 5} \right)4 + b$
$\Rightarrow \,\,\, - 1 = - 20 + b$
Solve for b
$b = - 1 + 20$
$\Rightarrow \,\,\,b = 19$
We can substitute for $m$ and $b$ in the equation (2) to find the equation of the line:
$\therefore\,\,\,y = - 5x + 19$

Hence, the equation of line that passes through point $\left( {4, - 1} \right)$ and has a slope of $m = - 5$ is $y = - 5x + 19$.

Note: To determine the equation of line we use the point-slope 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.