Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{10}{x+y} + \frac{2}{x-y} = 4, \\ \frac{15}{x+y} – \frac{9}{x-y} = -2 \]

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

\[ 10a + 2b = 4 \quad \text{(1)} \]

\[ 15a – 9b = -2 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 2b = 4 – 10a \]

\[ b = 2 – 5a \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 15a – 9(2 – 5a) = -2 \]

\[ 15a – 18 + 45a = -2 \]

\[ 60a = 16 \]

\[ a = \frac{4}{15} \]

Step 4: Find the Value of b

Substitute \(a = \frac{4}{15}\) into equation (3):

\[ b = 2 – 5\left(\frac{4}{15}\right) = 2 – \frac{4}{3} = \frac{2}{3} \]

Step 5: Find the Values of x and y

\[ x + y = \frac{1}{a} = \frac{15}{4},\quad x – y = \frac{1}{b} = \frac{3}{2} \]

Adding both equations:

\[ 2x = \frac{15}{4} + \frac{3}{2} = \frac{21}{4} \Rightarrow x = \frac{21}{8} \]

Substitute \(x = \frac{21}{8}\) into \(x + y = \frac{15}{4}\):

\[ y = \frac{15}{4} – \frac{21}{8} = \frac{9}{8} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{21}{8},\quad y = \frac{9}{8} \]

\[ \therefore \quad \text{The solution is } \left(\frac{21}{8},\; \frac{9}{8}\right). \]