Solve the following simultaneous equations y + 4x + 1 = 0, y^2 + 5x^2 + 2x = 0

Here we can either use the substitution method or the elimination method. The elimination method here would be hard to carry out due to there being no term which we can make equal in terms of only one of the variables in order to eliminate that variable. Therefore we choose substitution. 

y = -4x - 1 is substituted into the second equation.

(-4x-1)^2 + 5x^2 + 2x = 0 which is only in the form of x's so we can now solve to find the possible x values.

1 + 8x + 16x^2 + 5x^2 + 2x = 0                21x^2 + 10x + 1 = 0

Here we need to solve a quadratic. First we check if we can have x's with no coefficients. As we have 21x^2, we cant. But, we can notice that we need 2 numbers that multiply to give 1. These numbers must be either -1 and -1 or 1 and 1. As we need our x's to multiply to 21x^2, we need the factors of 21x^2. 21 and 2 or 7 and 3. 21 and 1 will be very far off ever giving 10x with the real parts only being 1 or -1. Therefore we try 7 and 3. 

(7x + 1)(3x + 1) is our solved quadratic.

x = -1/7, -1/3

Subbing the x values back into equation 1 gives our y values

y = -3/7, 1/3