Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

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

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

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

\[ 9a – 4b = 1 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 3a = 2 – 2b \]

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

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

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

\[ 3(2 – 2b) – 4b = 1 \]

\[ 6 – 6b – 4b = 1 \]

\[ 6 – 10b = 1 \]

\[ 10b = 5 \]

\[ b = \frac{1}{2} \]

Step 4: Find the Value of a

Substitute \(b = \frac{1}{2}\) into equation (3):

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

Step 5: Find the Values of x and y

\[ x + y = \frac{1}{a} = 3,\quad x – y = \frac{1}{b} = 2 \]

Adding both equations:

\[ 2x = 5 \Rightarrow x = \frac{5}{2} \]

\[ y = 3 – \frac{5}{2} = \frac{1}{2} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{5}{2},\quad y = \frac{1}{2} \]

\[ \therefore \quad \text{The solution is } \left(\frac{5}{2},\; \frac{1}{2}\right). \]