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# The sum of two numbers is 82. Their difference is 24. How do you write a system of equations that describes these situations and solve them by elimination to find the two numbers?

Last updated date: 29th Feb 2024
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Hint:In the given question, we have been asked to find the two numbers and it is given that the sum of those numbers is 82 while the difference is 24. In order to solve the question, first we need to assume the two numbers as variables and later we will make two equations according to the question.
And then solve the equations by elimination method, we will get our two numbers.

Complete step by step solution:
We have given that,
The sum of two numbers is 82.
Let assume the two numbers be $x$ and$y$.
Here we have to add $x$ and$y$, the given sum is equals to 82
Thus, we get the equation
$\Rightarrow x+y=82$----- (1)
Now,
It is given that the sum of two numbers is 24.
Thus, we get the equation
$\Rightarrow x-y=24$----- (2)
Now, we are having two equations;
$\Rightarrow x+y=82$----- (1)
$\Rightarrow x-y=24$----- (2)
In the question, it is given that we have to solve these equations by elimination. So for solving the equations by elimination we have to perform a mathematical equation between these two equations in a way such that one of the variables gets eliminated and we get the value of the other variable.
Now,
Adding the equation (1) and equation (2), we get
$\Rightarrow \left( x+y \right)+\left( x-y \right)=82+24$
Simplifying the above equation, we get
$\Rightarrow x+y+x-y=106$
In the above equation, we can see that ‘y’ got eliminated, we get
$\Rightarrow 2x=106$
Solving for ‘x’, we get
$\Rightarrow x=53$
Putting the value of ‘x’ is equals to 53 in equation (1),
$\Rightarrow 53+y=82$
Solving the above equation for ‘y’, we get
$\Rightarrow y=82-53=29$
$\Rightarrow y=29$
Therefore, the two numbers are 53 and 29.

Note:
While solving the question, students are required to write the digits and their respective sign very carefully. The chances of mistakes only happen when solving the equations. Students should always remember that if they get any question like this, whenever we find the values, we should start solving the question by assuming those values as variables.