Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{5}{x+y} – \frac{2}{x-y} = -1, \\ \frac{15}{x+y} + \frac{7}{x-y} = 10 \]

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

\[ 5a – 2b = -1 \quad \text{(1)} \]

\[ 15a + 7b = 10 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 5a = -1 + 2b \]

\[ a = \frac{-1 + 2b}{5} \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 15\left(\frac{-1 + 2b}{5}\right) + 7b = 10 \]

\[ 3(-1 + 2b) + 7b = 10 \]

\[ -3 + 6b + 7b = 10 \]

\[ 13b = 13 \]

\[ b = 1 \]

Step 4: Find the Value of a

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

\[ a = \frac{-1 + 2(1)}{5} = \frac{1}{5} \]

Step 5: Find the Values of x and y

\[ x + y = \frac{1}{a} = 5,\quad x – y = \frac{1}{b} = 1 \]

Adding both equations:

\[ 2x = 6 \Rightarrow x = 3 \]

\[ y = 5 – 3 = 2 \]

Conclusion

The solution of the given system of equations is:

\[ x = 3,\quad y = 2 \]

\[ \therefore \quad \text{The solution is } (3,\; 2). \]