Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{1}{5x} + \frac{1}{6y} = 12, \\ \frac{1}{3x} – \frac{3}{7y} = 8 \]

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

\[ \frac{a}{5} + \frac{b}{6} = 12 \quad \text{(1)} \]

\[ \frac{a}{3} – \frac{3b}{7} = 8 \quad \text{(2)} \]

Step 2: Remove Fractions

Multiply equation (1) by 30:

\[ 6a + 5b = 360 \quad \text{(3)} \]

Multiply equation (2) by 21:

\[ 7a – 9b = 168 \quad \text{(4)} \]

Step 3: Express One Variable in Terms of the Other

From equation (3):

\[ 5b = 360 – 6a \]

\[ b = \frac{360 – 6a}{5} \quad \text{(5)} \]

Step 4: Substitute in Equation (4)

Substitute equation (5) into equation (4):

\[ 7a – 9\left(\frac{360 – 6a}{5}\right) = 168 \]

Multiply both sides by 5:

\[ 35a – 3240 + 54a = 840 \]

\[ 89a = 4080 \]

\[ a = \frac{4080}{89} \]

Step 5: Find the Value of b

Substitute \(a = \frac{4080}{89}\) into equation (5):

\[ b = \frac{360 – 6\left(\frac{4080}{89}\right)}{5} = \frac{1512}{89} \]

Step 6: Find the Values of x and y

\[ x = \frac{1}{a} = \frac{89}{4080},\quad y = \frac{1}{b} = \frac{89}{1512} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{89}{4080},\quad y = \frac{89}{1512} \]

\[ \therefore \quad \text{The solution is } \left(\frac{89}{4080},\; \frac{89}{1512}\right). \]