Question

Xy+x+y=9 x^2y+xy^2=20

Answer 1

Решение///////////////////////////

Answer 2

To solve the system of equations:

1. xy + x + y = 9 .......(Equation 1)
2. x^2y + xy^2 = 20 .......(Equation 2)

We can use substitution or elimination. Let's use the substitution method:

Step 1: Solve Equation 1 for one variable in terms of the other. Let's solve for y:

xy + x + y = 9 y(x + 1) = 9 - x y = (9 - x) / (x + 1)

Step 2: Substitute the expression we found for y into Equation 2:

x^2y + xy^2 = 20 x^2[(9 - x) / (x + 1)] + x[(9 - x) / (x + 1)]^2 = 20

Step 3: Simplify the equation:

x^2(9 - x) + x(9 - x)^2 = 20(x + 1)

Step 4: Expand and rearrange the equation:

9x^2 - x^3 + x(81 - 18x + x^2) = 20x + 20 9x^2 - x^3 + 81x - 18x^2 + x^3 = 20x + 20

Step 5: Combine like terms:

9x^2 - 18x^2 + x^3 - x^3 - 20x = 20 - 81 -9x^2 - 20x + 61 = 0

Step 6: Now, we have a quadratic equation in terms of x. We can solve it using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:

The quadratic formula states that for an equation in the form of ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In our equation, a = -9, b = -20, and c = 61. Substituting these values into the quadratic formula:

x = (20 ± √((-20)^2 - 4(-9)(61))) / 2(-9)

Step 7: Calculate the discriminant:

Discriminant = b^2 - 4ac Discriminant = (-20)^2 - 4(-9)(61) Discriminant = 400 + 2208 Discriminant = 2608

Step 8: Since the discriminant (2608) is positive, there are two real solutions for x:

x = (20 + √2608) / -18 or x = (20 - √2608) / -18

Step 9: Now, calculate the two values of x:

x = (20 + √2608) / -18 ≈ -2.6936 or x = (20 - √2608) / -18 ≈ 2.3599

Step 10: Finally, substitute the values of x back into the expression we found for y:

For x ≈ -2.6936: y = (9 - (-2.6936)) / (-2.6936 + 1) ≈ 4.5263

For x ≈ 2.3599: y = (9 - 2.3599) / (2.3599 + 1) ≈ 2.1869

So, the two sets of solutions to the system of equations are approximately:

1. x ≈ -2.6936, y ≈ 4.5263
2. x ≈ 2.3599, y ≈ 2.1869