Need help with solve the system of equations x^3-y^2=7.75. And 3xy+y=3.5
well, the 2nd equation says
y = 3.5/(3x+1)
so, plugging that into the 1st equation, we have
x^3 - 12.25/(3x+1)^2 = 7.75
(36x^5+24x^4+4x^3-279x^2-186x-80) / 4(3x+1)^2 = 0
a little synthetic division yields x=2
To solve the system of equations x^3 - y^2 = 7.75 and 3xy + y = 3.5, we can use the method of substitution.
Step 1: Solve the second equation for y in terms of x.
In the second equation, we have 3xy + y = 3.5. Factoring out y, we get y(3x + 1) = 3.5. Dividing both sides by (3x + 1), we have y = 3.5 / (3x + 1).
Step 2: Substitute the value of y into the first equation.
In the first equation, we have x^3 - y^2 = 7.75. Substitute the value of y from step 1 into the equation, we get x^3 - (3.5 / (3x + 1))^2 = 7.75.
Simplifying the equation, we have x^3 - 3.5^2 / (3x + 1)^2 = 7.75.
Step 3: Solve the equation for x.
To solve the equation x^3 - 3.5^2 / (3x + 1)^2 = 7.75, we need to manipulate the equation, so it's easier to solve. Cross-multiply to get rid of the fraction: (3x + 1)^2 * (x^3 - 7.75) - 3.5^2 = 0.
Expand and simplify the equation: (3x + 1)^2 * x^3 - (3x + 1)^2 * 7.75 - 3.5^2 = 0.
Next, distribute and combine like terms: 9x^5 + 6x^4 + x^3 - 7.75 * (3x + 1)^2 - 12.25 = 0.
This is now a quintic equation, which can be challenging to solve algebraically. To find the exact solutions, we would need to use numerical methods or graphing calculators/software.
However, if you just need an approximate solution, you can use numerical methods like the Newton-Raphson method or technology such as calculators or online equation solvers. These methods can provide a numerical approximation for the values of x that satisfy the equation.
I hope this explains how to solve the system of equations x^3 - y^2 = 7.75 and 3xy + y = 3.5. Let me know if you have any further questions!