Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{1}{2x} + \frac{1}{3y} = 2, \\ \frac{1}{3x} + \frac{1}{2y} = \frac{13}{6} \]

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

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

\[ \frac{a}{3} + \frac{b}{2} = \frac{13}{6} \quad \text{(2)} \]

Step 2: Remove Fractions

Multiply equation (1) by 6:

\[ 3a + 2b = 12 \quad \text{(3)} \]

Multiply equation (2) by 6:

\[ 2a + 3b = 13 \quad \text{(4)} \]

Step 3: Express One Variable in Terms of the Other

From equation (3):

\[ 3a = 12 – 2b \]

\[ a = 4 – \frac{2b}{3} \quad \text{(5)} \]

Step 4: Substitute in Equation (4)

Substitute equation (5) into equation (4):

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

\[ 8 – \frac{4b}{3} + 3b = 13 \]

\[ 8 + \frac{5b}{3} = 13 \]

\[ \frac{5b}{3} = 5 \]

\[ b = 3 \]

Step 5: Find the Value of a

Substitute \(b = 3\) into equation (5):

\[ a = 4 – \frac{2(3)}{3} = 2 \]

Step 6: 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:

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

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