The system of nonlinear equations from the given statement and find the numbers

Tahmid Knox

Tahmid Knox

Answered question

2021-09-23

The system of nonlinear equations from the given statement and find the numbers involved in the problem by solving the system.

Answer & Explanation

pattererX

pattererX

Skilled2021-09-24Added 95 answers

Calculation:
It is given that the sum of two numbers is 20 and their product is 96.
Consider the first part of the sentence as "sum of two numbers is 20" and the second part as "product is 96".
Step 1: Assume the numbers to be x and y.
Step 2: Model the given conditions into a system of equations.
From the first part, the equation is modeled as x+y=20 and from the second part, the equation is xy=96.
Thus, the system of equations becomes {x+y=20xy=96
Step 3: Modify the equation xy=96 as y=96x.
Substitute y=96x in the equation x+y=20 and obtain the quadratic equation as,
x+(96x)=20
x2+96=20x (multiplied by x on both sides)
x220x+96=0
Solve the equation x220x+96=0 and obtain the value of x as follows.
x220x+96=0
x212x8x+96=0
x(x12)8(x6)=0
(x12)(x8)=0
On further simplification gives,
x12=0 or x80
x=12 or x=8
Substitute x=12 in the equation xy=96 and obtain the value of y as follows.
xy=96
(12)y=96
y=9612
y=8
Substitute x=8 in the equation xy=96 and obtain the value of y as follows.
xy=96
(8)y=96
y=968
y=12
Thus, for x=12,y=8, and for x=8,y=12.
Hence, the solution set is {(8,12),(12,8)}.
Step 4: Check the result, by substituting the obtained solutions in the given original equations x+y=20 and xy=96.
Substitute (8,12) in the given system and check.
(8)+(12)=20
20=20
(8)(12)=96
96=96
Substitute (12,8) in the given system and check.
(12)+(8)=20
20=20
(12)(8)=96
96=96
Therefore, the two numbers are 12 and 8 or 8 and 12.

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