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Answer: The two numbers are 151 and 66.
Step-by-step explanation: We are to find the two numbers whose sum is 217 and difference is 85.
Let, 'x' and 'y' represents the two numbers, where x > y.
Then, according to the given information, we have
[tex]x+y=217~~~~~~~~~~~~(i)\\\\x-y=85~~~~~~~~~~~~~(ii)[/tex]
Adding equations (i) and (ii), we get
[tex]x+x=217+85\\\\\Rightarrow 2x=302\\\\\Rightarrow x=\dfrac{302}{2}\\\\\Rightarrow x=151,[/tex]
and from equation (i), we get
[tex]y=217-x=217-151=66.[/tex]
Therefore, x = 151 and y = 66.
Thus, the two numbers are 151 and 66.
Considering the definition of a system of linear equations, the two numbers that have a sum of 217 and a difference of 85 are 66 and 151.
System of linear equations
A system of linear equations is a set of two or more equations of the first degree, in which two or more unknowns are related.
Solving a system of equations consists of finding the value of each unknown so that all the equations of the system are fulfilled. That is to say, the values of the unknowns must be sought, with which when replacing, they must give the solution proposed in both equations.
This case
In this case, a system of linear equations must be proposed taking into account that 'x' and 'y' represents the two numbers.
You know tha two numbers have a sum of 217 and a difference of 85. So, the system of equations to solve is:
[tex]\left \{ {{x+y=217} \atop {x-y=85}} \right. [/tex]
There are several methods to solve a system of equations, it is decided to solve it using the substitution method, which consists of clearing one of the two variables in one of the equations of the system and substituting its value in the other equation.
In this case, isolating the variable "x" from the second equation, you obtain:
x= 85 + y
Substituting in the first equation you get the expression:
85 + y +y= 217
Solving:
2y= 217 - 85
2y= 132
y= 132÷ 2
y= 66
Finally, to obtain the value of "x" it is replaced in the previously obtained expression for "x":
x= 85 + y
x= 85 + 66
x= 151
Finally, the two numbers that have a sum of 217 and a difference of 85 are 66 and 151.
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