To form an equation with the solution [tex]\( p = \frac{1}{5} \)[/tex], let's follow the steps in deriving it.
1. Form a Simple Linear Equation:
We start by considering a basic form of a linear equation:
[tex]\[
ap - b = 0
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( p \)[/tex] is the variable.
2. Choose Appropriate Values for Constants:
We need to choose values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that when [tex]\( p = \frac{1}{5} \)[/tex], the equation holds true. Let's choose [tex]\( a = 5 \)[/tex] and solve for [tex]\( b \)[/tex].
3. Plug in the Solution to Determine [tex]\( b \)[/tex]:
Substitute [tex]\( p = \frac{1}{5} \)[/tex] into the equation and solve for [tex]\( b \)[/tex]:
[tex]\[
5 \left( \frac{1}{5} \right) - b = 0
\][/tex]
Simplify the expression inside the parenthesis:
[tex]\[
1 - b = 0
\][/tex]
Therefore, we have:
[tex]\[
b = 1
\][/tex]
4. Construct the Final Equation:
Now that we have determined [tex]\( a = 5 \)[/tex] and [tex]\( b = 1 \)[/tex], plug these values back into the original form of the equation:
[tex]\[
5p - 1 = 0
\][/tex]
Thus, the equation that has the solution [tex]\( p = \frac{1}{5} \)[/tex] is:
[tex]\[
5p - 1 = 0
\][/tex]
This equation satisfies the requirement because substituting [tex]\( p = \frac{1}{5} \)[/tex] correctly balances the equation to zero.
