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Question: What is the equation of the line, \(y=\dfrac{5}{7}x-12\) in standard form?...

What is the equation of the line, y=57x−12y=\dfrac{5}{7}x-12 in standard form?

Explanation

Solution

In order to rewrite the given line expression in standard form, we need to transpose the term containing ‘x’ and the constant from R.H.S. to L.H.S., thus making the Right-Hand Side of the equation equal to zero. Now, we will rearrange the terms and compare it with the standard equation of a line to get a standardized line equation.

Complete step-by-step answer:
We have been given our line equation as a linear equation in two variables: y=57x−12y=\dfrac{5}{7}x-12 . This is a line equation in two dimensions.
Now, the standard form of writing a line equation in two dimensions is, ax+by+c=0ax+by+c=0, where, a, b and c are constants such that a and b cannot be simultaneously equal to zero.

Thus, to rewrite our equation in standard form, we need to transpose the term containing ‘x’ and the constant from R.H.S. to L.H.S. . We can do that in the following manner:
⇒y=57x−12 ⇒y−57x+12=0 \begin{aligned} & \Rightarrow y=\dfrac{5}{7}x-12 \\\ & \Rightarrow y-\dfrac{5}{7}x+12=0 \\\ \end{aligned}

Now, we can rationalize this equation by multiplying it with 7 on both sides of the equation. This is done as follows:
⇒(y−57x+12)×7=0×7 ⇒7y−5x+84=0 or,5x−7y−84=0 \begin{aligned} & \Rightarrow \left( y-\dfrac{5}{7}x+12 \right)\times 7=0\times 7 \\\ & \Rightarrow 7y-5x+84=0 \\\ & or,5x-7y-84=0 \\\ \end{aligned}

On comparing our above equation with the standard line equation, we get:
⇒a=5 ⇒b=−7 ⇒c=−84 \begin{aligned} & \Rightarrow a=5 \\\ & \Rightarrow b=-7 \\\ & \Rightarrow c=-84 \\\ \end{aligned}

Here, we can see that the constant terms a and b are not equal to zero simultaneously. Thus, the equation obtained is the correct equation.
Hence, the equation of the line, y=57x−12y=\dfrac{5}{7}x-12 in standard form can be written as 5x−7y−84=05x-7y-84=0.

Note: While transposing a term from L.H.S. to R.H.S. or vice versa, we should always make sure to change its sign or else or solution will be wrong. Also, the main aim of transposing terms from one side of the equation to another is to always bring like terms together and separately isolate the variable.