Question

The system of equations (1 + y^2 - y - 3x + frac(1)/(y) = 0) and (2y + 8 = 12x + 10 + 4xy + 7 = 3) has exactly one solution. Use substitution to find the solution. What is the solution? Enter your answer as an ordered pair, like this: (42, 53)

a) (4, 2)
b) (5, 6)
c) (1, 3)
d) (2, 5)

Answer 1

To solve the given system of equations using substitution, solve the first equation for y, then substitute this value of y into the second equation. Simplify the resulting equation and solve for x. Finally, substitute the value of x back into the first equation to solve for y. However, in this case, the given system of equations does not have exactly one solution.

Step-by-step explanation:

To solve the given system of equations using substitution, we'll start by solving the first equation for y:

1 + y^2 - y - 3x + \frac{1}{y} = 0

y^2 - y + \frac{1}{y} = 3x - 1

Next, we'll substitute this value of y into the second equation:

2y + 8 = 12x + 10 + 4xy + 7 - 3

2y = 12x + 14 + 4xy

Substituting the expression for y from the first equation:

2\left(3x - 1 + \frac{1}{3x - 1}\right) = 12x + 14 + 4x\left(3x - 1 + \frac{1}{3x - 1}\right)

Expanding and simplifying:

6x - 2 + \frac{2}{3x - 1} = 12x + 14 + 12x - 4 + \frac{4}{3x - 1}

8 = 18x + 12x + 12x - 6 - 4

8 = 42x - 10

42x = 18

x = \frac{3}{7}

Substituting this value of x back into the first equation to solve for y:

1 + y^2 - y - 3\left(\frac{3}{7}\right) + \frac{1}{y} = 0

Simplifying:

y^2 - y + \frac{1}{y} = \frac{9}{7} - 1

y^2 - y + \frac{1}{y} = \frac{2}{7}

Multiplying by y to clear the fraction:

y^3 - y^2 + 1 = \frac{2y}{7}

y^3 - y^2 - \frac{2y}{7} + 1 = 0

Unfortunately, we are unable to find an exact solution for y in this equation. Therefore, the system of equations does not have exactly one solution.