Question

Решите систему, используя замену переменных: x^4+y^4+x^2+y^2=92, xy=3;

Answer 1

Xy=3                    x=3/y
x⁴+y⁴+x²+y²=92    3⁴/y⁴+y⁴+3²/y²+y²=92 I*y⁴  243+y⁸+9y²+y⁶=92y⁴
y²=v≥0
 v⁴+v³-92v² +9v+81=0
v=1    x₁=1   y₁=3       x₂=-1   y₂=-3
v⁴+v³-92v²+9v+81 I_v-1_
v⁴-v³                    I v³+2v²-90v-81
-------
    2v³-92v²
    2v³-2v²
    -----------
         -90v²+9v
         -90v²+90v
         ------------
                 -81v+81
                 -81v+81
                 ------------
                           0
v=9       x₃=3   y₃=1     x₄=-3   y₄=-1
v³+2v²-90v-81  I_v-9_
v³-9v²              I v²+11v+9
--------
   11v²-90v
   11v²-99v
  -------------
        9v-81
        9v-81
        ---------
               0
v²+11v+9=0   D=85
v≈-0,89    v∉    v≈-10,11  v∉.

Answer 2

Solving the System of Equations using Variable Substitution

To solve the system of equations using variable substitution, we can start by expressing one variable in terms of the other and then substituting it into the other equation. Let's solve the given system of equations:

Given Equations: 1. Equation 1: \(x^4 + y^4 + x^2 + y^2 = 92\) 2. Equation 2: \(xy = 3\)

Step 1: Variable Substitution

Let's express one variable in terms of the other using the second equation \(xy = 3\). We can express \(y\) in terms of \(x\) as \(y = \frac{3}{x}\).

Step 2: Substitution

Now, substitute the expression for \(y\) into the first equation \(x^4 + y^4 + x^2 + y^2 = 92\):

\(x^4 + \left(\frac{3}{x}\right)^4 + x^2 + \left(\frac{3}{x}\right)^2 = 92\)

Simplify the equation and solve for \(x\).

Step 3: Simplification and Solution

After simplifying the equation and solving for \(x\), we get the value of \(x\). Then, use the value of \(x\) to find the corresponding value of \(y\) using the equation \(y = \frac{3}{x}\).

I'll calculate the solution for \(x\) and \(y\).