Question

If three numbers are added the sum is 15. If the second number is subtracted from the sum of the first and the third number then we get 5. If twice the first number is added to the second and third number is subtracted from the sum we get 4. Find the three numbers.

Answer 1

Hint: Start by letting the three unknown numbers as variables. In the question three situations are given, convert these situations from statement form to mathematical equations and solve the mathematical equations to get the answer to the above question. So we can start by assuming x, y, z as the three numbers and as per the condition in the question, we will get the first equation as \[x+y+z=5\] . Similarly, other equations can be formed and solved further.

Complete step-by-step answer:
To start with the question, we let the first number to be x, the second number to be y and the third number to be z.
Now it is given in the question that the sum of the three numbers is 15. These can be mathematically represented as:
\[x+y+z=15\ldots \ldots \ldots \ldots \ldots \ldots \ldots \left( i \right)\]
It is also given in the question that if the second number is subtracted from the sum of the first and the third number then we get 5. So, our equation for these statement can be:
\[x+z-y=5\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left( ii \right)\]
The third condition given in the question is: if twice the first number is added to the second and third number is subtracted from the sum we get 4.
\[2x+y-z=4\ldots \ldots \ldots \ldots \ldots \ldots ..\left( iii \right)\]
Now to solve the equations, first we will add equation (ii) and equation (iii). On doing so, we get
\[3x=9\]
 $ \Rightarrow x=3 $
Now we will subtract equation (ii) from equation (i). On doing so, we get
\[2y=10\]
 $ \Rightarrow y=5 $
Now we will substitute the value of x and y in equation (i). On doing so, we get
\[3+5+z=15\]
 $ \Rightarrow z=7 $
So, we can conclude that the three numbers are 3, 5 and 7.

Note: Whenever you are dealing with equations with 3 unknown variables, it is difficult to solve the equations and there is always a possibility that no solution occurs for the set of three equations. The best way to solve three equations is to use one equation to eliminate one of the three variables from the other two equations so that you are left with 2 equations in 2 variables, which is easier to solve. Cramer’s rule can also be used.