Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{7x-2y}{xy} = 5, \\ \frac{8x+7y}{xy} = 15 \]

Solution

Step 1: Simplify the Equations

First equation:

\[ \frac{7x}{xy} – \frac{2y}{xy} = 5 \]

\[ \frac{7}{y} – \frac{2}{x} = 5 \quad \text{(1)} \]

Second equation:

\[ \frac{8x}{xy} + \frac{7y}{xy} = 15 \]

\[ \frac{8}{y} + \frac{7}{x} = 15 \quad \text{(2)} \]

Step 2: Make Suitable Substitution

Let

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

Then equations (1) and (2) become:

\[ -2a + 7b = 5 \quad \text{(3)} \]

\[ 7a + 8b = 15 \quad \text{(4)} \]

Step 3: Express One Variable in Terms of the Other

From equation (3):

\[ 7b = 5 + 2a \]

\[ b = \frac{5 + 2a}{7} \quad \text{(5)} \]

Step 4: Substitute in Equation (4)

Substitute equation (5) into equation (4):

\[ 7a + 8\left(\frac{5 + 2a}{7}\right) = 15 \]

Multiply both sides by 7:

\[ 49a + 40 + 16a = 105 \]

\[ 65a = 65 \]

\[ a = 1 \]

Step 5: Find the Value of b

Substitute \(a = 1\) into equation (5):

\[ b = \frac{5 + 2(1)}{7} = 1 \]

Step 6: Find the Values of x and y

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

Conclusion

The solution of the given system of equations is:

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

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