Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{2}{x} + \frac{3}{y} = 13, \\ \frac{5}{x} – \frac{4}{y} = -2 \]

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

\[ 2a + 3b = 13 \quad \text{(1)} \]

\[ 5a – 4b = -2 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 2a = 13 – 3b \]

\[ a = \frac{13 – 3b}{2} \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 5\left(\frac{13 – 3b}{2}\right) – 4b = -2 \]

Multiply both sides by 2:

\[ 65 – 15b – 8b = -4 \]

\[ 65 – 23b = -4 \]

\[ 23b = 69 \]

\[ b = 3 \]

Step 4: Find the Value of a

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

\[ a = \frac{13 – 9}{2} = 2 \]

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:

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

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