Solve {l}{-54x+117=0}{text{Solvefor}ytext{where}}{y=x^4-6x^3+22x^2}

$$\left. \begin{array} { l } { -54 x + 117 = 0 }\\ { \text{Solve for } y \text{ where} } \\ { y = x ^ {4} - 6 x ^ {3} + 22 x ^ {2} } \end{array} \right.$$

Solve for x, y

$$y = \frac{83317}{1296} = 64\frac{373}{1296} \approx 64.287808642$$

Solution Steps

Consider the first equation. Subtract 117 from both sides. Anything subtracted from zero gives its negation.

$$-54x=-117$$

Divide both sides by -54.

$$x=\frac{-117}{-54}$$

Reduce the fraction \frac{-117}{-54} to lowest terms by extracting and canceling out -9.

$$x=\frac{13}{6}$$

Consider the second equation. Insert the known values of variables into the equation.

$$y=\left(\frac{13}{6}\right)^{4}-6\times \left(\frac{13}{6}\right)^{3}+22\times \left(\frac{13}{6}\right)^{2}$$

Calculate \frac{13}{6} to the power of 4 and get \frac{28561}{1296}.

$$y=\frac{28561}{1296}-6\times \left(\frac{13}{6}\right)^{3}+22\times \left(\frac{13}{6}\right)^{2}$$

Calculate \frac{13}{6} to the power of 3 and get \frac{2197}{216}.

$$y=\frac{28561}{1296}-6\times \frac{2197}{216}+22\times \left(\frac{13}{6}\right)^{2}$$

Multiply -6 and \frac{2197}{216} to get -\frac{2197}{36}.

$$y=\frac{28561}{1296}-\frac{2197}{36}+22\times \left(\frac{13}{6}\right)^{2}$$

Subtract \frac{2197}{36} from \frac{28561}{1296} to get -\frac{50531}{1296}.

$$y=-\frac{50531}{1296}+22\times \left(\frac{13}{6}\right)^{2}$$

Calculate \frac{13}{6} to the power of 2 and get \frac{169}{36}.

$$y=-\frac{50531}{1296}+22\times \frac{169}{36}$$

Multiply 22 and \frac{169}{36} to get \frac{1859}{18}.

$$y=-\frac{50531}{1296}+\frac{1859}{18}$$

Add -\frac{50531}{1296} and \frac{1859}{18} to get \frac{83317}{1296}.

$$y=\frac{83317}{1296}$$

The system is now solved.

$$x=\frac{13}{6} y=\frac{83317}{1296}$$