Question

How do you find the standard form for the line with (6,7); m= undefined?

Answer 1

Hint: The standard form of the equation of a straight line is \[ax+by+c=0\], here \[a,b\And c\in \] Real numbers. We can find the slope of the line using the coefficients of the equation. The slope of the line in terms of coefficients equals \[\dfrac{-a}{b}\].

Complete answer:
We are given that a line passes through points \[(6,7)\], and its slope is undefined. We have to find the standard form of the equation of this line. Let’s say that the standard form of the equation of line is \[ax+by+c=0\], we can find the values of \[a,b\And c\] using the given information, as follows
The slope of this line should be \[\dfrac{-a}{b}\], we are given that this value is undefined. A fraction is said to be undefined if its denominator is zero. Hence, here value of b is 0. substitute this value in the equation of line we assumed, we get
\[\begin{align}
  & \Rightarrow ax+0\times y+c=0 \\
 & \Rightarrow ax+c=0 \\
\end{align}\]
Dividing both sides by \[a\], we get
 \[\begin{align}
  & \Rightarrow ax\times \dfrac{1}{a}+c\times \dfrac{1}{a}=0\times \dfrac{1}{a} \\
 & \Rightarrow x+\dfrac{c}{a}=0 \\
\end{align}\]
We know that the line passes through the point \[(6,7)\]. So, it has to satisfy the equation of the line. Substituting this point in the above equation of line. We get,
\[\Rightarrow 6+\dfrac{c}{a}=0\]
Subtracting 6 from both sides of the equation, we get
\[\begin{align}
  & \Rightarrow 6+\dfrac{c}{a}-6=0-6 \\
 & \Rightarrow \dfrac{c}{a}=-6 \\
\end{align}\]
Substituting this value in the equation of line we made, we get
\[\Rightarrow x-6=0\]
Hence the standard form of the equation is \[x-6=0\].

Note: It should be remembered that if the slope of the line is undefined or zero, then in that case the equation of the line is \[x-a=0\] or \[y-a=0\] respectively. The value of \[a\] can be found using other information given about the line.