Question

A number exceeds the other by 26. If their sum is 54, find the numbers.

Answer 1

Hint: Assume that the numbers are x and y. So, the sum will be \[x+y=\text{ }54\] will be the first equation. Also, one number exceeds the other by 26, so \[y=\text{ }x+26\] will be the second equation, Now, we just need to solve these two linear equations in two variables to get the numbers x and y.

Complete step-by-step answer:
In the question, we have been given that a number exceeds the other by 26 and their sum is 54, so we have to find the two numbers. Now, two find the two numbers, let us take the two numbers to be x and y. Now, we need two linear equations, in order to solve for x and y. Here, assume that y is the bigger number than x.
So, to find the two equations, we will use two conditions given in the question. Here the first condition is given that a number exceeds the other by 26. So, the first equation that we will get will be: \[y=\text{ }x+26\] .
Now, the second condition is given that their sum is 54, so the second equation we will get is:
 \[x+y=\text{ }54\]
Now, for the two equations \[y=\text{ }x+26\] and \[x+y=\text{ }54\] , we will find x and y, as follows:
 \[\text{Subsititute}\;y=x+26\] in the equation \[x+y=54\] , to get:
 \[\begin{align}
  & \Rightarrow x+x+26=54 \\
 & \Rightarrow 2x+26=54 \\
 & \Rightarrow 2x=28 \\
 & \Rightarrow x=14 \\
\end{align}\]
So, now this value of x is substituted in the equation \[y=x+26\] , to get:
 \[\begin{align}
  & \Rightarrow y=x+26 \\
 & \Rightarrow y=14+26 \\
 & \Rightarrow y=40 \\
\end{align}\]
So, this means that here the two numbers are \[x=14\] and \[y=40\] .

Note: We have to be careful while solving creating two linear equations. Here, we can also put \[x=\text{ }y+26\] , if we assume that the number x is the bigger number than number y. Still that will give the same two numbers.