Find all solutions of the system of equations. (If there is no solution, enter NO SOLUTION.)
4x^2− 27y^3 = 35
8x^2+54y^3= 74
divide the second equation by 2
and the two equations are
4x^2 − 27y^3 = 35
4x^2 + 27y^3 = 37
--------------------- add
8 x^2 = 72
x^2 = 9
x = +/- 3
if x = + 3
36 -27 y^3 =35
-27 y^3 = -1
y^3 = 1/27
y = 1/3
so (3,1/3) would be a solution
same solution for y
which answer best describes the number of solutions for the following system of equations? 4x+2y = 5 8x+2y = -6 answers
To find the solutions of the given system of equations:
1. Let's set up the system of equations again for convenience:
4x^2 - 27y^3 = 35 .....(1)
8x^2 + 54y^3 = 74 .....(2)
2. To eliminate one variable, we'll multiply equation (1) by 2 and equation (2) by -1:
8x^2 - 54y^3 = 70 .....(3)
-8x^2 - 54y^3 = -74 .....(4)
3. Now, we'll add equations (3) and (4) to eliminate the x-variable:
(8x^2 - 54y^3) + (-8x^2 - 54y^3) = 70 - 74
-108y^3 = -4
4. Divide both sides of the equation by -108 to solve for y^3:
-108y^3 / -108 = -4 / -108
y^3 = 1/27
5. Now, let's take the cube root of both sides to solve for y:
∛(y^3) = ∛(1/27)
y = 1/3
6. Substitute the value of y into either equation (1) or (2) to solve for x. Let's use equation (1):
4x^2 - 27(1/3)^3 = 35
4x^2 - 27/27 = 35
4x^2 - 1 = 35
7. Simplify further by combining like terms:
4x^2 = 36
8. Divide both sides of the equation by 4 to solve for x:
4x^2 / 4 = 36 / 4
x^2 = 9
9. Take the square root of both sides to solve for x:
√(x^2) = √9
x = ±3
10. Therefore, the solutions to the given system of equations are:
x = 3, y = 1/3
x = -3, y = 1/3
So, there are two solutions to the system of equations.
To find the solutions of the given system of equations, we can use the method of substitution or elimination. Let's use substitution to solve this system.
From the first equation,
4x^2 - 27y^3 = 35 ...(Equation 1)
We can rearrange Equation 1 to solve for x^2 in terms of y:
4x^2 = 35 + 27y^3
x^2 = (35 + 27y^3)/4
Substituting this expression for x^2 into the second equation:
8((35 + 27y^3)/4) + 54y^3 = 74 ...(Equation 2)
Simplifying Equation 2:
2(35 + 27y^3) + 54y^3 = 74
70 + 54y^3 + 54y^3 = 74
108y^3 = 4
y^3 = 4/108
y^3 = 1/27
y = ∛(1/27)
y = 1/3
Substituting the value of y back into Equation 1:
4x^2 - 27(1/3)^3 = 35
4x^2 - 1/3 = 35
4x^2 = 35 + 1/3
4x^2 = 106/3
x^2 = (106/3) / 4
x^2 = 106/12
x^2 = 53/6
Taking the square root of both sides:
x = ± √(53/6)
Therefore, the solutions to the system of equations are:
x = √(53/6), y = 1/3
x = -√(53/6), y = 1/3