Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{4}{x} + 3y = 14, \\ \frac{3}{x} – 4y = 23 \]

Solution

Step 1: Make Suitable Substitution

Let

\[ \frac{1}{x} = a \]

Then the given equations become:

\[ 4a + 3y = 14 \quad \text{(1)} \]

\[ 3a – 4y = 23 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 3y = 14 – 4a \]

\[ y = \frac{14 – 4a}{3} \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 3a – 4\left(\frac{14 – 4a}{3}\right) = 23 \]

Multiply both sides by 3:

\[ 9a – 56 + 16a = 69 \]

\[ 25a = 125 \]

\[ a = 5 \]

Step 4: Find the Value of y

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

\[ y = \frac{14 – 20}{3} \]

\[ y = -2 \]

Step 5: Find the Value of x

\[ x = \frac{1}{a} = \frac{1}{5} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{1}{5},\quad y = -2 \]

\[ \therefore \quad \text{The solution is } \left(\frac{1}{5},\; -2\right). \]