Final answer:
To use the elimination method to find the solutions of the system of equations, multiply the equations by appropriate factors to make the x^2 coefficients the same. Then subtract the equations to eliminate x. Solve for y and substitute the values of x and y into one of the original equations to check for a solution. However, in this case, the system of equations has no solutions.
Step-by-step explanation:
To solve the system of equations using the elimination method, start by multiplying the first equation by 4 and the second equation by 3 to make the coefficients of the x^2 terms the same. This will allow you to eliminate the variable x by subtracting the two equations. Then, solve for y by substituting the value of x into one of the original equations. Finally, substitute the values of x and y into one of the equations to check if they satisfy the system.
Let's solve it step by step:
Multiplying the first equation by 4: 12x^2 - 4y^2 = 44
Multiplying the second equation by 3: 3x^2 + 12y^2 = 24
Subtracting the two equations to eliminate x: (12x^2 - 4y^2) - (3x^2 + 12y^2) = 44 - 24
Simplifying the equation: 9x^2 - 16y^2 = 20
Now, substitute the value of x^2 from this equation into the other equation to solve for y: x^2 = (8 - 4y^2) / 3
Substituting: 9(8 - 4y^2) / 3 - 16y^2 = 20
Simplifying the equation: 24 - 12y^2 - 16y^2 = 60
Combining the like terms: -28y^2 = 36
Dividing both sides by -28: y^2 = -1.29
Taking the square root of both sides: y = ±√(-1.29)
Since the square root of a negative number is not a real number, there are no solutions to this system of equations.