Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{3}{x} – \frac{1}{y} = -9, \\ \frac{2}{x} + \frac{3}{y} = 5 \]

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

\[ 3a – b = -9 \quad \text{(1)} \]

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

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ -b = -9 – 3a \]

\[ b = 3a + 9 \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 2a + 3(3a + 9) = 5 \]

\[ 2a + 9a + 27 = 5 \]

\[ 11a = -22 \]

\[ a = -2 \]

Step 4: Find the Value of b

Substitute \(a = -2\) into equation (3):

\[ b = 3(-2) + 9 = 3 \]

Step 5: Find the Values of x and y

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

Conclusion

The solution of the given system of equations is:

\[ x = -\frac{1}{2},\quad y = \frac{1}{3} \]

\[ \therefore \quad \text{The solution is } \left(-\frac{1}{2},\; \frac{1}{3}\right). \]