Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{5}{x-1} + \frac{1}{y-2} = 2, \\ \frac{6}{x-1} – \frac{3}{y-2} = 1 \]

Solution

Step 1: Make Suitable Substitution

Let

\[ \frac{1}{x-1} = a,\quad \frac{1}{y-2} = b \]

Then the given equations become:

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

\[ 6a – 3b = 1 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ b = 2 – 5a \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 6a – 3(2 – 5a) = 1 \]

\[ 6a – 6 + 15a = 1 \]

\[ 21a = 7 \]

\[ a = \frac{1}{3} \]

Step 4: Find the Value of b

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

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

Step 5: Find the Values of x and y

\[ x – 1 = \frac{1}{a} = 3 \Rightarrow x = 4 \]

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

Conclusion

The solution of the given system of equations is:

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

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