Solve System of 2 Equalities: 6x$^2$ + 19y + 3x + 15y$^2$ + 5y

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In summary: For $y=\frac{-1+\sqrt{3}i}{2}$, $x=\frac{-5(\frac{-1+\sqrt{3}i}{2})(3(\frac{-1+\sqrt{3}i}{2})+1)+19(\frac{-1+\sqrt{3}i}{2})x}{3(2x+1)}=\frac{-1+\sqrt{3}i}{6}$For $y=\frac{-1-\sqrt{3}i}{2}$, $x=\frac{-5(\frac{-1-\sqrt{3}i}{2})(3(\
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Solve the following system:

1) $6x^2+x(19y+3)+15y^2+5y=0$

2)$10x^3+(11y+5)x^2+(3y^2+3y+12)x+6y+6=0 $
 
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To solve this system, we can use the method of substitution.

First, we will solve equation 1 for x in terms of y. We can rewrite the equation as:

$6x^2+x(19y+3)+15y^2+5y=0$

$6x^2+x(19y+3)+15y^2+5y=0$

$6x^2+19xy+3x+15y^2+5y=0$

$6x^2+3x+19xy+15y^2+5y=0$

$3x(2x+1)+5y(3y+1)+19xy=0$

$3x(2x+1)=-5y(3y+1)-19xy$

$x=-\frac{5y(3y+1)+19xy}{3(2x+1)}$

Now, we can substitute this value of x into equation 2 and solve for y.

$10x^3+(11y+5)x^2+(3y^2+3y+12)x+6y+6=0$

$10\left(-\frac{5y(3y+1)+19xy}{3(2x+1)}\right)^3+(11y+5)\left(-\frac{5y(3y+1)+19xy}{3(2x+1)}\right)^2+(3y^2+3y+12)\left(-\frac{5y(3y+1)+19xy}{3(2x+1)}\right)+6y+6=0$

After simplifying, we get a single equation in terms of y:

$y^3+3y^2+2y+2=0$

Using the Rational Root Theorem, we can find that y=-2 is a root of this equation. Therefore, we can rewrite the equation as:

$(y+2)(y^2+y+1)=0$

Using the quadratic formula, we can find the other two roots to be $y=\frac{-1\pm\sqrt{3}i}{2}$.

Now, we can substitute these values of y back into the equation for x to find the corresponding values of x:

For $y=-2$, $x=\frac{-5(-2)(3(-2)+1
 

FAQ: Solve System of 2 Equalities: 6x$^2$ + 19y + 3x + 15y$^2$ + 5y

1. How do I solve this system of equations?

To solve this system of equations, you can use the substitution method or the elimination method. The goal is to get one variable isolated on one side of the equation, and then substitute that value into the other equation to solve for the other variable.

2. What are the steps for solving a system of equations?

The steps for solving a system of equations are:
1. Identify the variables in each equation
2. Choose a method (substitution or elimination)
3. Solve for one variable in one equation
4. Substitute that value into the other equation
5. Solve for the other variable
6. Check your solution by plugging it into both equations.

3. Can this system of equations have more than one solution?

Yes, this system of equations can have more than one solution. If the two equations represent two lines that intersect at more than one point, then there will be multiple solutions.

4. How can I check if my solution is correct?

You can check your solution by plugging it into both equations and seeing if it satisfies both equations. If it does, then your solution is correct.

5. Is there a specific order in which I should solve the equations?

No, there is no specific order in which you should solve the equations. You can choose to solve for either variable first, as long as you follow the steps for solving a system of equations.

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