Verified answer

Solve the second given equation for x:

x = 7 - y^2

Substitute this expression for x into the first given equation:

(7 - y^2)^2 - y = 7

Expand:

y^4 - 14y^2 + 49 - y = 7

Combine like terms and arrange in standard form:

y^4 - 14y^2 - y + 42 = 0

Use the rational roots theorem (maybe aided by looking at the graph) to

find the roots at y = -3, and y = 2

Use the factor theorem to convert these two roots into factors: (y + 3) (y - 2)

Divide the quartic function by the product of the known factors:

(y^4 - 14y^2 - y + 42) / ((y + 3) (y - 2)) = y^2 - y - 7

Solve this reduced quadratic using the quadratic formula:

y = (1 - √29)/2 and y = (1 + √29)/2

So altogether the four solutions for y are:

y = -3, y = 2, y = (1 - √29)/2 and y = (1 + √29)/2

Substitute these values for y, one at a time, into x + y^2 = 7 and evaluate x:

x = -2, x = 3, x = (√29 - 1)/2, and x = (-1-√29)/2

So the four solutions to the original problem are:

(-2, -3), (3, 2), ((√29 - 1)/2 , (1 - √29)/2) , and ((-1-√29)/2, (1 + √29)/2)