Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations, where \(y \ne 0\):

\[ 2x – \frac{3}{y} = 9, \\ 3x + \frac{7}{y} = 2 \]

Solution

Step 1: Express One Variable in Terms of the Other

From the first equation:

\[ 2x – \frac{3}{y} = 9 \]

\[ 2x = 9 + \frac{3}{y} \]

\[ x = \frac{9}{2} + \frac{3}{2y} \quad \text{(1)} \]

Step 2: Substitute in the Second Equation

Substitute equation (1) into the second equation:

\[ 3\left(\frac{9}{2} + \frac{3}{2y}\right) + \frac{7}{y} = 2 \]

\[ \frac{27}{2} + \frac{9}{2y} + \frac{7}{y} = 2 \]

\[ \frac{27}{2} + \frac{23}{2y} = 2 \]

Multiply both sides by 2:

\[ 27 + \frac{23}{y} = 4 \]

\[ \frac{23}{y} = -23 \]

\[ \frac{1}{y} = -1 \]

\[ y = -1 \]

Step 3: Find the Value of x

Substitute \(y = -1\) into equation (1):

\[ x = \frac{9}{2} + \frac{3}{2(-1)} \]

\[ x = \frac{9}{2} – \frac{3}{2} = 3 \]

Conclusion

The solution of the given system of equations is:

\[ x = 3,\quad y = -1 \]

\[ \therefore \quad \text{The solution is } (3,\; -1). \]