Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{15}{u} + \frac{2}{v} = 17, \\ \frac{1}{u} + \frac{1}{v} = \frac{36}{5} \]

Solution

Step 1: Make Suitable Substitution

Let

\[ \frac{1}{u} = a,\quad \frac{1}{v} = b \]

Then the given equations become:

\[ 15a + 2b = 17 \quad \text{(1)} \]

\[ a + b = \frac{36}{5} \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (2):

\[ b = \frac{36}{5} – a \quad \text{(3)} \]

Step 3: Substitute in Equation (1)

Substitute equation (3) into equation (1):

\[ 15a + 2\left(\frac{36}{5} – a\right) = 17 \]

\[ 15a + \frac{72}{5} – 2a = 17 \]

\[ 13a + \frac{72}{5} = 17 \]

\[ 13a = 17 – \frac{72}{5} = \frac{13}{5} \]

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

Step 4: Find the Value of b

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

\[ b = \frac{36}{5} – \frac{1}{5} = \frac{35}{5} = 7 \]

Step 5: Find the Values of u and v

\[ u = \frac{1}{a} = 5,\quad v = \frac{1}{b} = \frac{1}{7} \]

Conclusion

The solution of the given system of equations is:

\[ u = 5,\quad v = \frac{1}{7} \]

\[ \therefore \quad \text{The solution is } \left(5,\; \frac{1}{7}\right). \]