Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{4}{x} + 5y = 7, \\ \frac{3}{x} + 4y = 5 \]

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

\[ 4a + 5y = 7 \quad \text{(1)} \]

\[ 3a + 4y = 5 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (2):

\[ 4y = 5 – 3a \]

\[ y = \frac{5 – 3a}{4} \quad \text{(3)} \]

Step 3: Substitute in Equation (1)

Substitute equation (3) into equation (1):

\[ 4a + 5\left(\frac{5 – 3a}{4}\right) = 7 \]

Multiply both sides by 4:

\[ 16a + 25 – 15a = 28 \]

\[ a + 25 = 28 \]

\[ a = 3 \]

Step 4: Find the Value of y

Substitute \(a = 3\) into equation (3):

\[ y = \frac{5 – 9}{4} = -1 \]

Step 5: Find the Value of x

\[ x = \frac{1}{a} = \frac{1}{3} \]

Conclusion

The solution of the given system of equations is:

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

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