testing wrote:Find all real solutions of the system:
x - y = 3
x^3 - y^3 = 387
Non-linear systems tend to be ugly, no matter what. But you might be able to simplify a bit by factoring
:. . . . .
Plug "3" in for the "x - y", and simplify by dividing through by the 3. Then solve the resulting literal equation
for, say, y in terms of x, using the Quadratic Formula
:. . . . .. . . . .. . . . .. . . . .
This gives you two solutions, presumably one from the "plus" and the other from the "minus" (which you can see on your calculator, if you graph Y1=X-3 and Y2=(X^3-387)^(1/3) in the same window). The solutions will be the places where a "half" above crosses the other line, y = x - 3, assuming there is a crossing. The solution to one "half" might start like this:. . . . .. . . . .. . . . .
This is a radical equation; you begin to solve
by squaring both sides:. . . . .. . . . .. . . . .. . . . .. . . . .
Finish the simplification, and then verify that each x-value is "allowed" inside the square root in the equation with the "
" above. Once you've found which, if any, of the values is allowable, back-solve (using "y = x - 3") for the corresponding y-values for that "half".
Then note that, due to the squaring, solving the other "half" should look very similar.