Solve the System of Equations by the Substitution Method

Video Explanation

Question

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

\[ x + y = 5xy, \\ x + 2y = 13xy \]

Solution

Step 1: Divide Each Equation by \(xy\)

From the first equation:

\[ \frac{x}{xy} + \frac{y}{xy} = 5 \]

\[ \frac{1}{y} + \frac{1}{x} = 5 \quad \text{(1)} \]

From the second equation:

\[ \frac{x}{xy} + \frac{2y}{xy} = 13 \]

\[ \frac{1}{y} + \frac{2}{x} = 13 \quad \text{(2)} \]

Step 2: Make Suitable Substitution

Let

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

Then equations (1) and (2) become:

\[ a + b = 5 \quad \text{(3)} \]

\[ 2a + b = 13 \quad \text{(4)} \]

Step 3: Express One Variable in Terms of the Other

From equation (3):

\[ b = 5 – a \quad \text{(5)} \]

Step 4: Substitute in Equation (4)

Substitute equation (5) into equation (4):

\[ 2a + (5 – a) = 13 \]

\[ a + 5 = 13 \]

\[ a = 8 \]

Step 5: Find the Value of b

Substitute \(a = 8\) into equation (5):

\[ b = 5 – 8 = -3 \]

Step 6: Find the Values of x and y

\[ x = \frac{1}{a} = \frac{1}{8},\quad y = \frac{1}{b} = -\frac{1}{3} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{1}{8},\quad y = -\frac{1}{3} \]

\[ \therefore \quad \text{The solution is } \left(\frac{1}{8},\; -\frac{1}{3}\right). \]