Let's solve the given system of equations using the substitution method step by step. We have the following equations:
[tex]\[
\sqrt{2} x + \sqrt{3} y = 0 \quad \text{(i)}
\][/tex]
[tex]\[
\sqrt{3} x - \sqrt{8} y = 0 \quad \text{(ii)}
\][/tex]
### Step 1: Solve one of the equations for one variable
Let's solve equation (i) for [tex]\( x \)[/tex]:
[tex]\[
\sqrt{2} x + \sqrt{3} y = 0
\][/tex]
[tex]\[
\sqrt{2} x = -\sqrt{3} y
\][/tex]
[tex]\[
x = -\frac{\sqrt{3}}{\sqrt{2}} y
\][/tex]
### Step 2: Substitute the expression for [tex]\( x \)[/tex] into the other equation
Now substitute [tex]\( x = -\frac{\sqrt{3}}{\sqrt{2}} y \)[/tex] into equation (ii):
[tex]\[
\sqrt{3}\left( -\frac{\sqrt{3}}{\sqrt{2}} y \right) - \sqrt{8} y = 0
\][/tex]
### Step 3: Simplify and solve for [tex]\( y \)[/tex]
Simplify the equation:
[tex]\[
-\frac{\sqrt{3} \cdot \sqrt{3}}{\sqrt{2}} y - \sqrt{8} y = 0
\][/tex]
[tex]\[
-\frac{3}{\sqrt{2}} y - 2\sqrt{2}y = 0
\][/tex]
To combine the terms, we need a common denominator. Note that [tex]\(2\sqrt{2}\)[/tex] can be written as [tex]\( \frac{4\sqrt{2}}{2} \)[/tex]:
[tex]\[
-\frac{3}{\sqrt{2}} y - \frac{4\sqrt{2}}{\sqrt{2}} y = 0
\][/tex]
[tex]\[
-\frac{3 + 4\sqrt{2}}{\sqrt{2}} y = 0
\][/tex]
Since [tex]\(\frac{3 + 4\sqrt{2}}{\sqrt{2}}\)[/tex] is a non-zero constant,
[tex]\[
y = 0
\][/tex]
### Step 4: Substitute [tex]\( y = 0 \)[/tex] back into the expression for [tex]\( x \)[/tex]
Using [tex]\( y = 0 \)[/tex] in [tex]\( x = -\frac{\sqrt{3}}{\sqrt{2}} y \)[/tex]:
[tex]\[
x = -\frac{\sqrt{3}}{\sqrt{2}} \cdot 0
\][/tex]
[tex]\[
x = 0
\][/tex]
### Solution
The solution to the system of equations is:
[tex]\[
x = 0, \quad y = 0
\][/tex]
This means the only solution that satisfies both equations is [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex].
