Solve the simultaneous equations: y + 4x + 1 = 0, and y^2 + 5x^2 + 2x = 0.

"Solve the simultaneous equations" is another way of saying find the unknown variables. In this case, they are x and y. When looking at the second equation, y^2+5x^2+2x=0, we can see that we have two unknown variables: x and y. We will proceed to find one of these, and use the answer to find the second. To find the first variable, we will use first equation, y+4x+1=0, rearranging it in such a way that it can be inserted into the second equation.
Rearranging equation 1 in terms of x (i.e. rearrange with x terms on one side, and y terms on the other), we get:y+4x+1=0---> subtract 4x+1 from both sides. y=-4x-1
Inserting this into the second equation in place of 'y'. (-4x-1)^2 + 5x^2 + 2x = 0---> Expanding the brackets. 16x^2 + 8x +1 + 5x^2 + 2x =0---> Grouping together similar terms (i.e. all the terms with x^2 in, we will add together etc.)21x^2 + 10x +1 = 0
Now you will notice that we have a quadratic equation. In order to find the value of x, we must factorise this. ---> (7x+1) (3x+1) = 0
Since we know the two brackets multiply together to equal zero, we know that one of the brackets must have a value of zero (since anything multiplied by zero equals zero). We take a bracket at a time, and make it equal zero. Then we solve the simple linear equation for x.
7x+1 = 0---> Rearrange for xx = -1/7
3x+1 = 0---> Rearrange for xx = -1/3
Now that we have found the values of x, these can be substituted back into one of the original equations un order to find y. ---> Substitute each value one at a time.
When x = -1/7:y + 4x + 1 = 0y + 4(-1/7) + 1 = 0y - 4/7 + 1 = 0y = -3/7
And when x = -1/3:y + 4x + 1 = 0y + 4(-1/3) + 1 = 0y - 4/3 + 1 = 0y = 1/3
Overall, this means that the solutions can be found. This is done by grouping the pairs of values.(-1/7, -3/7)(-1/3, 1/3)