k i need to solve the following system:
7x - 8y = 24
xy^2 = 1
i figured out that x = 1/y^2 and i can reduce it to :
7=24y^2+8y^3
but i don't know how to reduce it anymore after that
I think I would solve it graphically, that is, plot on my graphing calculator the two original equations, and see where they intersect
Graphically solving the system of equations is indeed a good approach to visually determine the point of intersection between the two equations. Here's how you can do it step by step:
1. Start by graphing the first equation, 7x - 8y = 24, on your graphing calculator or a graphing software. To do this, rearrange the equation to solve for y:
y = (7x - 24) / 8
2. Next, graph the second equation, xy^2 = 1. Rewrite it as:
y^2 = 1 / x
3. Now, choose a range of x-values to plot both equations. You can pick a range that you think will include the point of intersection.
4. Plot the first equation on your graphing calculator or software by substituting various x-values into the equation and determining the corresponding y-values. Connect the points to form a line.
5. Plot the second equation by substituting various x-values into the equation and calculating the corresponding y-values. Connect the points to form a curve.
6. Look for the point where the line and the curve intersect. This point represents the solution to the system of equations (x, y).
If you have access to a graphing calculator or a graphing software, following these steps should help you find the approximate solution to the system of equations.