Question

How do you write the standard form of equation given\[\left( {3,5} \right)\] and slope\[\dfrac{5}{3}\] ?

Answer 1

Hint:The standard form for linear equations in two variables is Ax+By=C where A, B and C are real numbers and both A and B are non-zero. In general, the slope intercept form formula is $y = mx + b$ where, $m$ is the slope and $b$ is the y-intercept. We need to find the standard form of the equation of a line through the given point with a given slope.

Complete step by step answer:
Given points are \[({x_1},{y_1}) = (3,5)\]with the slope\[m = \dfrac{5}{3}\]. Here, we will use the slope point formula as we are given a point and a slope of the line.
\[{y_2} - {y_1} = m({x_2} - {x_1})\]
Let, \[({x_2},{y_2}) = (x,y)\]
Substituting the values in the above formula, we get,
\[ \Rightarrow y - 5 = \frac{5}{3}(x - 3)\]
Taking LCM as \[3\]on both the side, we get,
\[ \Rightarrow y(3) - 5(3) = 5(x - 3)\]
Removing the brackets, we get,
\[ \Rightarrow 3y - 15 = 5x - 15\]
By using transposing in this above equation, we get,
\[ \Rightarrow 3y - 5x = - 15 + 15\]
\[ \Rightarrow 3y - 5x = 0\]
Again by using transposition, we move all from LHS to RHS, we get,
\[ \Rightarrow 0 = + 5x - 3y\]
Rearranging the above equation, we get,
\[ \Rightarrow 5x - 3y = 0\]
Thus, the equation in slope intercept form is :
\[5x - 3y = 0\]
\[ \Rightarrow 5x = 3y\]
\[ \Rightarrow 3y = 5x\]
\[ \therefore y = \dfrac{5}{3}x\]

Hence, the above equation can be rewritten in the standard form as \[ - 5x + 3y = 0\] for the given point \[\left( {3,5} \right)\] and slope \[\dfrac{5}{3}\].

Note:The slope-intercept form is given as $y=mx+b$ where $m$ is the slope and b is the y-intercept when $x=0$ or at point (0,b). Since we used the coordinates of one known point and the slope to write this form of equation it is called the point-slope form. Also, since the slope of a vertical line is undefined you can't write the equation of a vertical line using either the slope-intercept form or the point-slope form. But you can express it using the standard form. We can also use the graph to show the line.