Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{2}{\sqrt{x}} – \frac{3}{\sqrt{y}} = 2, \\ \frac{4}{\sqrt{x}} – \frac{9}{\sqrt{y}} = -1 \]

Solution

Step 1: Make Suitable Substitution

Let

\[ \frac{1}{\sqrt{x}} = a,\quad \frac{1}{\sqrt{y}} = b \]

Then the given equations become:

\[ 2a – 3b = 2 \quad \text{(1)} \]

\[ 4a – 9b = -1 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 2a = 2 + 3b \]

\[ a = 1 + \frac{3}{2}b \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 4\left(1 + \frac{3}{2}b\right) – 9b = -1 \]

\[ 4 + 6b – 9b = -1 \]

\[ 4 – 3b = -1 \]

\[ 3b = 5 \]

\[ b = \frac{5}{3} \]

Step 4: Find the Value of a

Substitute \(b = \frac{5}{3}\) into equation (3):

\[ a = 1 + \frac{3}{2}\left(\frac{5}{3}\right) \]

\[ a = 1 + \frac{5}{2} = \frac{7}{2} \]

Step 5: Find the Values of x and y

\[ \sqrt{x} = \frac{1}{a} = \frac{2}{7} \Rightarrow x = \left(\frac{2}{7}\right)^2 = \frac{4}{49} \]

\[ \sqrt{y} = \frac{1}{b} = \frac{3}{5} \Rightarrow y = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{4}{49},\quad y = \frac{9}{25} \]

\[ \therefore \quad \text{The solution is } \left(\frac{4}{49},\; \frac{9}{25}\right). \]