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How do you find a standard form equation for the line \[y=\dfrac{2}{3}x-7\]?

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Last updated date: 20th Jul 2024
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Answer
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Hint: This type of problem is based on the concept of equation of lines. First, we have to consider the given equation. Subtract \[\dfrac{2}{3}x\] from both the sides of the equation and convert the given equation in such a way that the variables x and y are in the left-hand side and constant is in the right-hand side. Then, multiply the whole obtained equation of line by 3. We get an equation of the form Ax+By=C, where A is the coefficient of x and B is the coefficient of y. Here, C is the constant. Thus, we have obtained the standard form of the line.

Complete step by step answer:
According to the question, we are asked to find the standard line equation for the line \[y=\dfrac{2}{3}x-7\].
We have been given the line equation is \[y=\dfrac{2}{3}x-7\]. ------(1)
First, we have to convert the entire variable to the left-hand side and constants to the right hand side of the equation.
Here, x is in the right-hand side of the equation (1).
We have to subtract \[\dfrac{2}{3}x\] from both the sides of the equation (1).
\[\Rightarrow y-\dfrac{2}{3}x=\dfrac{2}{3}x-\dfrac{2}{3}x-7\]
We know that the terms with the same magnitude and opposite signs cancel out.
Therefore, we get
\[\Rightarrow y-\dfrac{2}{3}x=-7\]
Since the coefficient f x is in fraction, we can convert the coefficient of x to an integer by multiplying the whole equation by 3.
\[\Rightarrow 3\left( y-\dfrac{2}{3}x \right)=-7\times 3\]
On further simplification, we get
\[\Rightarrow 3\left( y-\dfrac{2}{3}x \right)=-21\]
Let us use the distributive property, that is, \[\left( a+b \right)c=ac+bc\], we get
\[3y-\dfrac{2}{3}\times 3x=-21\]
On cancelling out the common term 3 from the numerator and denominator from the x coefficient, we get
\[3y-2x=-21\]
Let us take -1 common from the LHS and RHS, we get
\[-1\left( 2x-3y \right)=-1\times 21\]
Cancelling out -1 from both the left-hand side and right-hand side of the equation, we get
\[\therefore 2x-3y=21\] ------(2)
We know that the standard form of a line equation is Ax+By=C, where A is the coefficient of x, B is the coefficient of y and C is a constant.
Comparing the standard form of a line equation with equation (2), we find that \[2x-3y=21\] is the standard form of a line.
Therefore, the standard form equation for the line \[y=\dfrac{2}{3}x-7\] is \[2x-3y=21\].

Note: We should not get confused with the slope-intercept form, point-intercept form and the standard form of a line. We should always make necessary calculations in the given equation to convert all the variables to the LHS and constants to the RHS. Also avoid calculation mistakes based on sign conventions. We can also keep the coefficients of x and y in fraction.