Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{2}{3x+2y} + \frac{3}{3x-2y} = \frac{17}{5}, \\ \frac{5}{3x+2y} + \frac{1}{3x-2y} = 2 \]

Solution

Step 1: Make Suitable Substitution

Let

\[ \frac{1}{3x+2y} = a,\quad \frac{1}{3x-2y} = b \]

Then the given equations become:

\[ 2a + 3b = \frac{17}{5} \quad \text{(1)} \]

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

Step 2: Express One Variable in Terms of the Other

From equation (2):

\[ b = 2 – 5a \quad \text{(3)} \]

Step 3: Substitute in Equation (1)

Substitute equation (3) into equation (1):

\[ 2a + 3(2 – 5a) = \frac{17}{5} \]

\[ 2a + 6 – 15a = \frac{17}{5} \]

\[ -13a + 6 = \frac{17}{5} \]

\[ -13a = \frac{17}{5} – \frac{30}{5} = -\frac{13}{5} \]

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

Step 4: Find the Value of b

Substitute \(a = \frac{1}{5}\) into equation (3):

\[ b = 2 – 5\left(\frac{1}{5}\right) = 1 \]

Step 5: Find the Values of x and y

\[ 3x + 2y = \frac{1}{a} = 5,\quad 3x – 2y = \frac{1}{b} = 1 \]

Adding both equations:

\[ 6x = 6 \Rightarrow x = 1 \]

Substitute \(x = 1\) into \(3x + 2y = 5\):

\[ 3 + 2y = 5 \Rightarrow 2y = 2 \Rightarrow y = 1 \]

Conclusion

The solution of the given system of equations is:

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

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