Question

If the difference of the two numbers is \[8\]and the difference of their square is\[160\]. What are the numbers?

Answer 1

Hint: Here in this above question we will start by taking the two numbers as \[x\]and\[y\]. Now by using the statement given in the question that is the difference between two numbers is \[8\] we will frame the first equation and by the use of the second statement that is the difference of their square is\[160\]we will frame the second equation. Later on by simplifying we will get the required result.

Formula used:
 In order to solve the above question we will firstly form the equation as per given statement and then by the use of substitution method we will get the desired results.

Complete step-by-step answer:
We will take two numbers as \[x\]and\[y\].
So the given first and second equations will be \[x - y = 8\]and \[{x^2} - {y^2} = 160\]respectively.
Now by simplifying the firstly equation we will take out value of \[x\]
\[x = y + 8\]
Later on put the value of \[x\]in the second equation we get
\[
   \Rightarrow {(y + 8)^2} - {y^2} = 160 \\
   \Rightarrow {y^2} + 16y + 64 - {y^2} = 160 \\
 \]
Now we will further simplify the above equation obtained by cancelling out \[{y^2}\]and \[ - {y^2}\]as their sum will be zero. So we get
\[
   \Rightarrow 16y + 64 = 160 \\
   \Rightarrow 16y = 160 - 64 \\
   \Rightarrow 16y = 96 \\
   \Rightarrow y = 6 \\
 \]
Now as we have obtained value of \[y\]so we will put it in the first equation and we get
\[
   \Rightarrow x - 6 = 8 \\
   \Rightarrow x = 8 + 6 \\
   \Rightarrow x = 14 \\
 \]
It means the two numbers will be \[14\]and \[6\]respectively

Note: While solving such types of problems we need to solve the linear equations in two variables. Such questions can be solved through two types of method substitution method and elimination method where to get the equation in one variable we either add or subtract the equations. Be careful while framing the equations as per the statements given to get desired results and avoid calculation mistakes.