Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{4}{x} + 3y = 8, \\ \frac{6}{x} – 4y = -5 \]

Solution

Step 1: Express One Variable in Terms of the Other

From the first equation:

\[ \frac{4}{x} + 3y = 8 \]

\[ 3y = 8 – \frac{4}{x} \]

\[ y = \frac{8}{3} – \frac{4}{3x} \quad \text{(1)} \]

Step 2: Substitute in the Second Equation

Substitute equation (1) into the second equation:

\[ \frac{6}{x} – 4\left(\frac{8}{3} – \frac{4}{3x}\right) = -5 \]

\[ \frac{6}{x} – \frac{32}{3} + \frac{16}{3x} = -5 \]

Combine the terms containing \( \frac{1}{x} \):

\[ \frac{18 + 16}{3x} – \frac{32}{3} = -5 \]

\[ \frac{34}{3x} – \frac{32}{3} = -5 \]

Multiply both sides by 3:

\[ \frac{34}{x} – 32 = -15 \]

\[ \frac{34}{x} = 17 \]

\[ x = 2 \]

Step 3: Find the Value of y

Substitute \(x = 2\) into equation (1):

\[ y = \frac{8}{3} – \frac{4}{3(2)} \]

\[ y = \frac{8}{3} – \frac{2}{3} = \frac{6}{3} = 2 \]

Conclusion

The solution of the given system of equations is:

\[ x = 2,\quad y = 2 \]

\[ \therefore \quad \text{The solution is } (2,\; 2). \]