Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{1}{7x} + \frac{1}{6y} = 3, \\ \frac{1}{2x} – \frac{1}{3y} = 5 \]

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

\[ \frac{a}{7} + \frac{b}{6} = 3 \quad \text{(1)} \]

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

Step 2: Remove Fractions

Multiply equation (1) by 42:

\[ 6a + 7b = 126 \quad \text{(3)} \]

Multiply equation (2) by 6:

\[ 3a – 2b = 30 \quad \text{(4)} \]

Step 3: Express One Variable in Terms of the Other

From equation (4):

\[ 3a = 30 + 2b \]

\[ a = 10 + \frac{2b}{3} \quad \text{(5)} \]

Step 4: Substitute in Equation (3)

Substitute equation (5) into equation (3):

\[ 6\left(10 + \frac{2b}{3}\right) + 7b = 126 \]

\[ 60 + 4b + 7b = 126 \]

\[ 11b = 66 \]

\[ b = 6 \]

Step 5: Find the Value of a

Substitute \(b = 6\) into equation (5):

\[ a = 10 + \frac{2(6)}{3} \]

\[ a = 14 \]

Step 6: Find the Values of x and y

\[ x = \frac{1}{a} = \frac{1}{14},\quad y = \frac{1}{b} = \frac{1}{6} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{1}{14},\quad y = \frac{1}{6} \]

\[ \therefore \quad \text{The solution is } \left(\frac{1}{14},\; \frac{1}{6}\right). \]