Question

How do you write an equation in slope – intercept form of the line with parametric equation: x = 2 + 3t and y = 4 + t?

Answer 1

We will first eliminate t from the two equations x = 2 + 3t and y = 4 + t to obtain the equation of line in x and y form. Then we will write it in the form y = mx + c which is the slope – intercept form.

Complete step by step answer:
We are given that we are required to write an equation in slope – intercept form of the line with parametric equation: x = 2 + 3t and y = 4 + t.
Let us assume x = 2 + 3t to be the first equation and y = 4 + t to be the second equation.
Multiplying the second equation by 3, we will then obtain the following equation:-
$\Rightarrow$ 3y = 12 + 3t …………….(3)
Subtracting equation number 3 from equation number 1, we will then obtain the following equation with us:-
$\Rightarrow$ {x} – {3y} = {2 + 3t} – {12 + 3t}
Removing the parentheses from both the sides of the above mentioned equation, we will then obtain the following equation with us:-
$\Rightarrow$ x – 3y = 2 + 3t – 12 – 3t
Simplifying the terms by clubbing like terms on the right hand side of above equation, we will then obtain the following equation with us:-
$\Rightarrow$ x – 3y = 2 - 12
Simplifying the terms by clubbing like terms on the right hand side of above equation further, we will then obtain the following equation with us:-
$\Rightarrow$ x – 3y = - 10
Taking x from addition in the left hand side to subtraction in the right hand side, we will then obtain the following equation:-
$\Rightarrow$– 3y = - 10 – x
Dividing both the sides of above equation by 3, we will then obtain:-
$\Rightarrow - y = - \dfrac{{10}}{3} - \dfrac{x}{3}$
Multiplying the above equation by – 1, we will then obtain the following equation with us:-
$\Rightarrow y = \dfrac{x}{3} + \dfrac{{10}}{3}$
Thus, we have the required equation.

Note: The students must note that the slope of the line is the tangent of the angle, the line makes with the positive direction of x – axis.
We convert an equation in slope – intercept form so as we can easily find the slope of the line without even lifting the hand and similarly the y – intercept of the equation.
It is easier to graph the line when it is in slope – intercept form as well.