Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations, where \(x \ne -1\) and \(y \ne 1\):

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

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

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

\[ 10a + 2b = \frac{5}{2} \quad \text{(2)} \]

Step 2: Remove Fractions

Multiply both equations by 2:

\[ 10a – 4b = 1 \quad \text{(3)} \]

\[ 20a + 4b = 5 \quad \text{(4)} \]

Step 3: Express One Variable in Terms of the Other

From equation (3):

\[ 10a = 1 + 4b \]

\[ a = \frac{1 + 4b}{10} \quad \text{(5)} \]

Step 4: Substitute in Equation (4)

Substitute equation (5) into equation (4):

\[ 20\left(\frac{1 + 4b}{10}\right) + 4b = 5 \]

\[ 2(1 + 4b) + 4b = 5 \]

\[ 2 + 8b + 4b = 5 \]

\[ 12b = 3 \]

\[ b = \frac{1}{4} \]

Step 5: Find the Value of a

Substitute \(b = \frac{1}{4}\) into equation (5):

\[ a = \frac{1 + 4\left(\frac{1}{4}\right)}{10} = \frac{2}{10} = \frac{1}{5} \]

Step 6: Find the Values of x and y

\[ x + 1 = \frac{1}{a} = 5 \Rightarrow x = 4 \]

\[ y – 1 = \frac{1}{b} = 4 \Rightarrow y = 5 \]

Conclusion

The solution of the given system of equations is:

\[ x = 4,\quad y = 5 \]

\[ \therefore \quad \text{The solution is } (4,\; 5). \]