Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \sqrt{2}x – \sqrt{3}y = 0, \\ \sqrt{3}x – \sqrt{8}y = 0 \]

Solution

Step 1: Express One Variable in Terms of the Other

From the first equation:

\[ \sqrt{2}x – \sqrt{3}y = 0 \]

\[ \sqrt{2}x = \sqrt{3}y \]

\[ x = \frac{\sqrt{3}}{\sqrt{2}}\,y \quad \text{(1)} \]

Step 2: Substitute in the Second Equation

Substitute equation (1) into the second equation:

\[ \sqrt{3}\left(\frac{\sqrt{3}}{\sqrt{2}}\,y\right) – \sqrt{8}y = 0 \]

\[ \frac{3}{\sqrt{2}}\,y – \sqrt{8}y = 0 \]

Since \(\sqrt{8} = 2\sqrt{2}\),

\[ \frac{3}{\sqrt{2}}\,y – 2\sqrt{2}\,y = 0 \]

\[ \left(\frac{3 – 4}{\sqrt{2}}\right)y = 0 \]

\[ -\frac{1}{\sqrt{2}}\,y = 0 \]

\[ y = 0 \]

Step 3: Find the Value of x

Substitute \(y = 0\) into equation (1):

\[ x = \frac{\sqrt{3}}{\sqrt{2}} \cdot 0 \]

\[ x = 0 \]

Conclusion

The solution of the given system of equations is:

\[ x = 0,\quad y = 0 \]

\[ \therefore \quad \text{The solution is } (0,\;0). \]