Question

How do you write $y = \dfrac{2}{3}x + 4$ in standard form?

Answer 1

Here, we will first write the standard form for a linear polynomial in general. Then we will apply the mathematical operation on the given equation. Finally, we will simplify the equation further to get the required standard form. A linear equation is an equation that has the highest degree of 1 and has one solution.

Complete step by step solution:
The linear equation given to us is $y = \dfrac{2}{3}x + 4$.
As we can see that the above equation is in slope-intercept form which is also a standard form but we have to find another standard form of this equation.
Another standard form is $Ax + By = C$, where $A,B,C$ are any constant terms.
Let us subtract $y$ from both side of equation, we get
$\begin{array}{l}y - y = - y + \dfrac{2}{3}x + 4\\\ \Rightarrow 0 = - y + \dfrac{2}{3}x + 4\end{array}$
Now, taking the L.C.M on RHS, we get
$\begin{array}{l} \Rightarrow 0 = \dfrac{{ - 3y + 2x + 4 \times 3}}{3}\\\ \Rightarrow 0 = \dfrac{{ - 3y + 2x + 12}}{3}\end{array}$
Multiplying both sides by 3, we get
$\Rightarrow - 3y + 2x + 12 = 0$
Taking the constant term on the right side, we get
$\Rightarrow - 3y + 2x = 12$
Rearranging the terms, we get
$\Rightarrow 2x - 3y = 12$

So another standard form of $y = \dfrac{2}{3}x + 4$ is $2x - 3y = 12$

Note:
The given equation is an equation of line in slope intercept form. The constant terms in the standard form are not unique as we can change them by multiplying or dividing the equation by any constant term. If the line is horizontal then $A = 0$ and also it cannot be a positive integer. The type of standard form used in the solution is also known as the general form of an equation. A Line is a one-dimensional figure that extends endlessly in both directions. It is also described as the shortest distance between any two points. There are many ways to form a line depending on the nature of the equation such as Point-slope Form, Intercept Form, Determinant Form and many others.