Question

A system of equations consists of y=x^3+5x+1 and y=x

Answer 1

x = (2^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) - 10 3^(1/3))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)) or x = -((-1)^(1/3) (2^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) + 10 (-3)^(1/3)))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)) or x = ((-1)^(1/3) ((-2)^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) + 10 3^(1/3)))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))

Explanation:

Solve for x:

y = x^3 + 5 x + 1

y = x^3 + 5 x + 1 is equivalent to x^3 + 5 x + 1 = y:

x^3 + 5 x + 1 = y

Subtract y from both sides:

1 + 5 x + x^3 - y = 0

Change coordinates by substituting x = z + λ/z, where λ is a constant value that will be determined later:

1 - y + 5 (z + λ/z) + (z + λ/z)^3 = 0

Multiply both sides by z^3 and collect in terms of z:

z^6 + z^4 (3 λ + 5) + z^3 (1 - y) + z^2 (3 λ^2 + 5 λ) + λ^3 = 0

Substitute λ = -5/3 and then u = z^3, yielding a quadratic equation in the variable u:

-125/27 + u^2 - u (y - 1) = 0

Find the positive solution to the quadratic equation:

u = 1/18 (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))

Substitute back for u = z^3:

z^3 = 1/18 (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))

Taking cube roots gives (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)/(2^(1/3) 3^(2/3)) times the third roots of unity:

z = (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)/(2^(1/3) 3^(2/3)) or z = -((-1/2)^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))/3^(2/3) or z = ((-1)^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))/(2^(1/3) 3^(2/3))

Substitute each value of z into x = z - 5/(3 z):

x = (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)/(2^(1/3) 3^(2/3)) - (5 (2/3)^(1/3))/(-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3) or x = -(5 (-1)^(2/3) (2/3)^(1/3))/(-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3) - (((-1)/2)^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))/3^(2/3) or x = (5 ((-2)/3)^(1/3))/(-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3) + ((-1)^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))/(2^(1/3) 3^(2/3))

Bring each solution to a common denominator and simplify:

Answer: x = (2^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) - 10 3^(1/3))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)) or x = -((-1)^(1/3) (2^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) + 10 (-3)^(1/3)))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3)) or x = ((-1)^(1/3) ((-2)^(1/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(2/3) + 10 3^(1/3)))/(6^(2/3) (-9 + 9 y + sqrt(3) sqrt(27 y^2 - 54 y + 527))^(1/3))

Answer 2

x=−0.24626617, y=−0.24626617

Explanation:

Substitute x3+5x+1 for y into y=x then solve for x

.

Replace y with x3+5x+1 in the equation.

x3+5x+1=x

Solve the equation for x

.

Remove parentheses.

x3+5x+1=x

Graph each side of the equation. The solution is the x-value of the point of intersection.

x≈−0.24626617

Substitute −0.24626617

for x into y=x then solve for y

.

Replace x with −0.24626617 in the equation.

x=−0.24626617

y=−0.24626617

Remove parentheses.

y=−0.24626617

The solution to the system is the complete set of ordered pairs that are valid solutions.

(−0.24626617,−0.24626617)

The result can be shown in multiple forms:

Point Form:

(−0.24626617,−0.24626617)

Equation Form:

x=−0.24626617, y=−0.24626617