Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations, where \(x \ne 0,\; y \ne 0\):

\[ \frac{2}{x} + \frac{3}{y} = \frac{9}{xy}, \\ \frac{4}{x} + \frac{9}{y} = \frac{21}{xy} \]

Solution

Step 1: Remove Denominators

Multiply both equations by \(xy\):

\[ 2y + 3x = 9 \quad \text{(1)} \]

\[ 4y + 9x = 21 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 2y = 9 – 3x \]

\[ y = \frac{9 – 3x}{2} \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 4\left(\frac{9 – 3x}{2}\right) + 9x = 21 \]

\[ 2(9 – 3x) + 9x = 21 \]

\[ 18 – 6x + 9x = 21 \]

\[ 3x = 3 \]

\[ x = 1 \]

Step 4: Find the Value of y

Substitute \(x = 1\) into equation (3):

\[ y = \frac{9 – 3(1)}{2} \]

\[ y = \frac{6}{2} = 3 \]

Conclusion

The solution of the given system of equations is:

\[ x = 1,\quad y = 3 \]

\[ \therefore \quad \text{The solution is } (1,\; 3). \]