Question

How do you write the standard form of the equation through:$(3,1)$, perpendicular to $y = - \dfrac{2}{3}x + 4$?

Answer 1

According to the question we have to determine the standard form of the equation through:$(3,1)$, perpendicular to $y = - \dfrac{2}{3}x + 4$. So, first of all to determine the required standard form of the equation we have to obtain the slope of the given expression which can be obtained with the help of the slope intercept form of the expression which is as mentioned below:
Formula used:
$\Rightarrow y - mx + c...............(A)$
Where, m is the slope of the expression and b is the y intercept.
Now, as mentioned in the question that the is perpendicular to the given expression which is $y = - \dfrac{2}{3}x + 4$as mentioned in the question so we have to determine the slope for the perpendicular line which can be determined with the help of the formula as given below:
Formula used:
$\Rightarrow$Perpendicular slope$m = - \dfrac{1}{m}..............(B)$
Now, to find the equation of the new line we have to use the points $(3,1)$which are as given in the question with the help of the formula as mentioned below:
Formula used:
$\Rightarrow y - {y_1} = m(x - {x_1})...............(C)$
Where, $({x_1},{y_1})$are points which are already given in the question are$(3,1)$and m is the perpendicular slope.
Now, we just have to solve the expression by multiplying/dividing or adding and subtracting the terms which can be.

Complete step by step solution:
Step 1: first of all to determine the required standard form of the equation we have to obtain the slope of the given expression which can be obtained with the help of the slope intercept form (A) of the expression which is as mentioned in the solution hint. Hence,
$\Rightarrow m = - \dfrac{2}{3}$
Step 2: Now, as mentioned in the question that the is perpendicular to the given expression which is $y = - \dfrac{2}{3}x + 4$as mentioned in the question so we have to determine the slope for the perpendicular line which can be determined with the help of the formula (B) as given in the solution. Hence,
$\Rightarrow$Perpendicular slope
$\Rightarrow m = - \dfrac{1}{{\dfrac{{ - 2}}{3}}} \\\ \Rightarrow m = \dfrac{3}{2} \\\$
Step 3: Now, to find the equation of the new line we have to use the points $(3,1)$which are as given in the question with the help of the formula (C) as mentioned in the solution hint. Hence,
$\Rightarrow y - 1 = \dfrac{3}{2}(x - 3)$
Step 4: Now, we just have to solve the expression as obtained in the solution step 3 by multiplying/dividing or adding and subtracting the terms which can be. Hence,
$\Rightarrow 2(y - 1) = 3(x - 3) \\\ \Rightarrow 2y - 2 = 3x - 9 \\\ \Rightarrow 3x - 2y = 7 \\\$

Hence, with the help of the formula (A), (B), and (C) we have determined the required standard form of the equation through:$(3,1)$, perpendicular to $y = - \dfrac{2}{3}x + 4$ which is $3x - 2y = 7$.

Note:

1. To determine the required standard form of the equation it is necessary that we have to determine the perpendicular slope for the line which can be determined with the help of the formula as mentioned in the solution hint.
2. It in line of equation$y - {y_1} = m(x - {x_1})$ points $({x_1},{y_1})$are the points of the equation through:$(3,1)$, perpendicular to $y = - \dfrac{2}{3}x + 4$.