Question

How to find the slope that is perpendicular to the line \[9x-y=9\]?

Answer 1

Hint: In order to solve the solve question, first we need to convert the given linear equation in the standard slope intercept form of a linear equation by simplifying the given equation. The slope intercept form of a linear equation is, \[y=mx+b\], where ‘m’ is the slope of the line. Then equating the product of two slopes equal to -1 and simplify for the value of slope of other line as if two lines are perpendicular then the product of their slope is equal to -1, i.e. \[{{m}_{1}}\times {{m}_{2}}=-1\]. In this way we will get the slope of the line that is perpendicular to the given line.

Complete step by step solution:
We have given that,
\[9x-y=9\]
As we know that the slope intercept form of a linear equation is,
\[y=mx+b\], where ‘m’ is the slope of the line and ‘b’ is the y-intercept.
Converting the given equation in slope intercept form of equation;
\[9x-y=9\]
Adding ‘y’ to both the sides of the equation, we get
\[9x=9+y\]
Subtracting 9 from both the sides of the equation, we get
\[9x-9=9+y-9\]
Simplifying the numbers in the above equation, we get
\[9x-9=y\]
Rewrite the above equation as;
\[y=9x-9\]
Comparing it with the slope intercept form of a linear equation i.e. \[y=mx+b\]
Thus,
Slope = m = \[9\]
We know that,
If two lines are perpendicular then the product of their slope is equal to -1, i.e. \[{{m}_{1}}\times {{m}_{2}}=-1\]
Therefore,
\[{{m}_{1}}\times {{m}_{2}}=-1\]
\[9\times {{m}_{2}}=-1\]
\[{{m}_{2}}=-\dfrac{1}{9}\]

Therefore, the slope of the line that is perpendicular to the line \[9x-y=9\] is \[-\dfrac{1}{9}\].

Note: While solving these types of questions, students need to know the concept of slope intercept form of linear equation and the relation between the slopes of two perpendicular lines. Solve the equation very carefully and do the calculation part very explicitly to avoid making any errors. For a straight line, if two points $A({{x}_{1}},{{y}_{1}})$ and $B({{x}_{2}},{{y}_{2}})$ are situated on the line, then by using the slope formula we can calculate the slope (m) as, $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.