Question

Write the equation of the line with point $\left( {3, - 3} \right)$ and slope 3.

Answer 1

Hint : The equation of the line is typically written as $y = mx + b$ where “m” is the slope and “b” is the y-intercept that is the value where the line cuts the y-axis. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables.

Complete step-by-step answer :
We begin by writing in the form of $y = mx + b$ hence we have,
$y = 3x + b$
The points or the coordinates are given in the question that is $\left( {3, - 3} \right)$ . These points represent “x” and “y” intercepts or the coordinates.
Now plug in the values for “x” and “y” from the point $\left( {3, - 3} \right)$ we have,
$- 3 = 3 \times 3 + b$
Solving this we have,
$\Rightarrow - 3 = 9 + b$
Further solving to simplest form,

$$\Rightarrow b = - 9 - 3 \\\ \Rightarrow b = - 12 \;$$

So, the value of “b” is $- 12$ . Now the equation in the slope intercept form is
$y = 3x - 12$
Hence the equation of the line with a slope of $3$ and passing through the point $\left( {3, - 3} \right)$ is $y = 3x - 12$ .
So, the correct answer is “ $y = 3x - 12$ ”.

Note : If we are given the slope and one point then these values can be substituted into a formula which is based on the definition for slope that is $y - {y_1} = m\left( {x - {x_1}} \right)$ . Here ${y_1}$ value will be $- 3$ , $m$ will be $3$ and ${x_1}$ value will be $3$ . On simplifying this answer , the equation of the line with a slope will be the same $y = 3x - 12$ . Therefore this is an alternative method which can be used to find the equation of the line when slope is given. The points always denote the “x” and “y” intercepts or the coordinates.