QuestionSolve for $x+y$ given the equation $x+3 \sqrt{y}=1+3 \sqrt{2}$ .

Studdy Solution

STEP 1

Assumptions1. We have the equation $x+3 \sqrt{y}=1+3 \sqrt{}$ . . We need to find the value of $x+y$ .

STEP 2

First, we need to isolate $x$ in the given equation. We can do this by subtracting $\sqrt{y}$ from both sides of the equation.
$x =1+ \sqrt{2} - \sqrt{y}$

STEP 3

Now, we need to find a way to express $y$ in terms of $x$ . To do this, we can square both sides of the equation to eliminate the square root.
$(x)^2 = (1+3 \sqrt{2} -3 \sqrt{y})^2$

STEP 4

Expand the right side of the equation.
$x^2 = (1+3 \sqrt{2})^2 -2(1+3 \sqrt{2})(3 \sqrt{y}) + (3 \sqrt{y})^2$

STEP 5

implify the right side of the equation.
$x^2 =1 +\sqrt{2} +18 -(1+3 \sqrt{2})\sqrt{y} +9y$

STEP 6

Combine like terms on the right side of the equation.
$x^2 =19 +6\sqrt{2} -6(1+3 \sqrt{2})\sqrt{y} +9y$

STEP 7

Now, we can isolate $y$ in the equation. First, subtract $19 +6\sqrt{2}$ from both sides of the equation.
$x^2 -19 -6\sqrt{2} = -6(1+3 \sqrt{2})\sqrt{y} +9y$

STEP 8

Next, add $6(1+3 \sqrt{2})\sqrt{y}$ to both sides of the equation.
$x^2 -19 -6\sqrt{2} +6(1+3 \sqrt{2})\sqrt{y} =y$

STEP 9

Finally, divide both sides of the equation by9 to solve for $y$ .
$y = \frac{x^2 -19 -6\sqrt{2} +6(+3 \sqrt{2})\sqrt{y}}{9}$

STEP 10

Now that we have expressions for both $x$ and $y$ , we can substitute these into the equation $x+y$ to find the value of this expression.
$x+y = (+3 \sqrt{2} -3 \sqrt{y}) + \frac{x^2 -19 -6\sqrt{2} +6(+3 \sqrt{2})\sqrt{y}}{9}$

STEP 11

implify the equation.
$x+y = \frac{9(+3 \sqrt{} -3 \sqrt{y}) + x^ -19 -6\sqrt{} +6(+3 \sqrt{})\sqrt{y}}{9}$

STEP 12

Further simplify the equation.
$x+y = \frac{9 +27\sqrt{2} -27\sqrt{y} + x^2 -19 -6\sqrt{2} +6\sqrt{2} +18\sqrt{2y}}{9}$

STEP 13

Combine like terms.
$x+y = \frac{x^2 -10 +27\sqrt{2} -27\sqrt{y} +24\sqrt{2y}}{9}$

STEP 14

implify the equation to find the value of $x+y$ .
$x+y = \frac{x^2 -10}{9} +3\sqrt{2} -3\sqrt{y} + \frac{8}{3}\sqrt{2y}$ The value of $x+y$ is $\frac{x^2 -10}{9} +3\sqrt{2} -3\sqrt{y} + \frac{8}{3}\sqrt{2y}$ .

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