Solve the System of Equations by the Substitution Method

Video Explanation

Question

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

\[ x + y = 2xy, \\ \frac{x – y}{xy} = 6 \]

Solution

Step 1: Simplify the Second Equation

\[ \frac{x – y}{xy} = 6 \]

\[ \frac{x}{xy} – \frac{y}{xy} = 6 \]

\[ \frac{1}{y} – \frac{1}{x} = 6 \quad \text{(1)} \]

Step 2: Simplify the First Equation

\[ x + y = 2xy \]

Divide both sides by \(xy\):

\[ \frac{x}{xy} + \frac{y}{xy} = 2 \]

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

Step 3: Make Suitable Substitution

Let

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

Then equations (1) and (2) become:

\[ b – a = 6 \quad \text{(3)} \]

\[ a + b = 2 \quad \text{(4)} \]

Step 4: Solve the Equations

From equation (4):

\[ b = 2 – a \quad \text{(5)} \]

Substitute equation (5) into equation (3):

\[ (2 – a) – a = 6 \]

\[ 2 – 2a = 6 \]

\[ -2a = 4 \]

\[ a = -2 \]

Step 5: Find the Value of b

Substitute \(a = -2\) into equation (5):

\[ b = 2 – (-2) = 4 \]

Step 6: Find the Values of x and y

\[ x = \frac{1}{a} = -\frac{1}{2},\quad y = \frac{1}{b} = \frac{1}{4} \]

Conclusion

The solution of the given system of equations is:

\[ x = -\frac{1}{2},\quad y = \frac{1}{4} \]

\[ \therefore \quad \text{The solution is } \left(-\frac{1}{2},\; \frac{1}{4}\right). \]