Solve for \(x\) in \(x+3y=162\).
Solve for \(x\).
\[x+3y=162\]
Subtract \(3y\) from both sides.
\[x=162-3y\]
\[x=162-3y\]
Substitute \(x=162-3y\) into \(\sqrt{2x}+2y=234\).
Start with the original equation.
\[\sqrt{2x}+2y=234\]
Let \(x=162-3y\).
\[\sqrt{2(162-3y)}+2y=234\]
Simplify.
\[\sqrt{6(54-y)}+2y=234\]
\[\sqrt{6(54-y)}+2y=234\]
Solve for \(y\) in \(\sqrt{6(54-y)}+2y=234\).
Solve for \(y\).
\[\sqrt{6(54-y)}+2y=234\]
Separate terms with roots from terms without roots.
\[\sqrt{6(54-y)}=234-2y\]
Square both sides.
\[6(54-y)=54756-936y+4{y}^{2}\]
Expand.
\[324-6y=54756-936y+4{y}^{2}\]
Move all terms to one side.
\[324-6y-54756+936y-4{y}^{2}=0\]
Simplify \(324-6y-54756+936y-4{y}^{2}\) to \(-54432+930y-4{y}^{2}\).
\[-54432+930y-4{y}^{2}=0\]
Use the Quadratic Formula.
In general, given \(a{x}^{2}+bx+c=0\), there exists two solutions where:
\[x=\frac{-b+\sqrt{{b}^{2}-4ac}}{2a},\frac{-b-\sqrt{{b}^{2}-4ac}}{2a}\]
In this case, \(a=-4\), \(b=930\) and \(c=-54432\).
\[{y}^{}=\frac{-930+\sqrt{{930}^{2}-4\times -4\times -54432}}{2\times -4},\frac{-930-\sqrt{{930}^{2}-4\times -4\times -54432}}{2\times -4}\]
Simplify.
\[y=\frac{-930+6\sqrt{167}\imath }{-8},\frac{-930-6\sqrt{167}\imath }{-8}\]
\[y=\frac{-930+6\sqrt{167}\imath }{-8},\frac{-930-6\sqrt{167}\imath }{-8}\]
Simplify solutions.
\[y=\frac{3(155-\sqrt{167}\imath )}{4},\frac{3(155+\sqrt{167}\imath )}{4}\]
\[y=\frac{3(155-\sqrt{167}\imath )}{4},\frac{3(155+\sqrt{167}\imath )}{4}\]
Substitute \(y=\frac{3(155-\sqrt{167}\imath )}{4},\frac{3(155+\sqrt{167}\imath )}{4}\) into \(x=162-3y\).
Start with the original equation.
\[x=162-3y\]
Let \(y=\frac{3(155-\sqrt{167}\imath )}{4},\frac{3(155+\sqrt{167}\imath )}{4}\).
\[x=162-3\times \frac{3(155-\sqrt{167}\imath )}{4},162-3\times \frac{3(155+\sqrt{167}\imath )}{4}\]
Simplify.
\[x=162-\frac{9(155-\sqrt{167}\imath )}{4},162-\frac{9(155+\sqrt{167}\imath )}{4}\]
\[x=162-\frac{9(155-\sqrt{167}\imath )}{4},162-\frac{9(155+\sqrt{167}\imath )}{4}\]
Therefore,
\[\begin{aligned}&x=162-\frac{9(155-\sqrt{167}\imath )}{4},162-\frac{9(155+\sqrt{167}\imath )}{4}\\&y=\frac{3(155-\sqrt{167}\imath )}{4},\frac{3(155+\sqrt{167}\imath )}{4}\end{aligned}\]
x=162-(9*(155-sqrt(167)*IM))/4,162-(9*(155+sqrt(167)*IM))/4;y=(3*(155-sqrt(167)*IM))/4,(3*(155+sqrt(167)*IM))/4
