Question

How do you solve \[\dfrac{9}{k-7}=\dfrac{6}{k}?\]

Answer 1

Hint:We are given an equation as \[\dfrac{9}{k-7}=\dfrac{6}{k}\] and we will first understand the type of equation, the degree of the equation and then we will understand what the solution means and then we will use algebraic tools first to simplify our equation into a linear form. Then we will take variables to one side and constant terms to the other side and simplify to get our answer.

Complete step by step solution:
We are given \[\dfrac{9}{k-7}=\dfrac{6}{k}.\] We can see that they have one variable k and each has just power as 1 only. So, they are linear and we will first simplify to get the exact knowledge of the equation. So, we will cross multiply the equation. We will multiply k – 7 to the right and k to the left and hence the equation \[\dfrac{9}{k-7}=\dfrac{6}{k}\] will become
\[\Rightarrow 9\left( k \right)=6\left( k-7 \right)\]
On opening the brackets, we get,
\[\Rightarrow 9k=6k-42\]
Now as we can see clearly that they are linear equations. So, they have only one solution.
Therefore, we will use algebraic tools to solve such problems easily. Now, we have
\[9k=6k-42\]
So, we will subtract 6k on both the sides, so we get,
\[\Rightarrow 9k-6k=6k-42-6k\]
On further simplifying, we get,
\[\Rightarrow 3k=-42\]
Now, dividing both the sides by 3, so we get,
\[\Rightarrow k=\dfrac{-42}{3}=-14\]
So, k = – 14 is the solution.

Note: We also have an alternate method to solve that is hit and trial method.
We will use the hit and trial method. In this method, we will put the value of k by assumption and check if it satisfies our equation or not.
So, we put k = 5 in 9k = 6k – 42. So, we get,
\[9\times 5=6\times 5-42\]
On simplifying, we get,
\[\Rightarrow 45=-12\]
So, k = 5 is not the solution.
We put k = 0 in 9k = 6k – 42. So, we get,
\[9\times 0=6\times 0-42\]
So, we get,
\[\Rightarrow 0=-42\]
So, k = 0 is not the solution.
We can see that the difference on the right side is decreasing. So, we are moving on the right side.
Now we put k = – 10 in 9k = 6k – 42, so we get,
\[9\times \left( -10 \right)=-6\times \left( -10 \right)-42\]
So, we get,
\[\Rightarrow -90=-102\]
So, k = – 10 is not our solution.
So, we put now k = – 14 in 9k = 6k – 42. So, we get,
\[9\left( -14 \right)=6\left( -14 \right)-42\]
On simplifying, we get,
\[\Rightarrow -126=-126\]
This is true and k = – 14 is the solution.
The hit and trial method is a bit long and it may be incorrect if we move along the wrong line. So, we will use algebraic tools to solve such problems easily. Also, remember when we divide the negative term by a positive solution, it is always negative.