Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{2}{x} + \frac{5}{y} = 1, \\ \frac{60}{x} + \frac{40}{y} = 19 \]

Solution

Step 1: Make Suitable Substitution

Let

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

Then the given equations become:

\[ 2a + 5b = 1 \quad \text{(1)} \]

\[ 60a + 40b = 19 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 2a = 1 – 5b \]

\[ a = \frac{1 – 5b}{2} \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 60\left(\frac{1 – 5b}{2}\right) + 40b = 19 \]

\[ 30 – 150b + 40b = 19 \]

\[ 30 – 110b = 19 \]

\[ 110b = 11 \]

\[ b = \frac{1}{10} \]

Step 4: Find the Value of a

Substitute \(b = \frac{1}{10}\) into equation (3):

\[ a = \frac{1 – \frac{5}{10}}{2} \]

\[ a = \frac{\frac{1}{2}}{2} = \frac{1}{4} \]

Step 5: Find the Values of x and y

\[ x = \frac{1}{a} = 4,\quad y = \frac{1}{b} = 10 \]

Conclusion

The solution of the given system of equations is:

\[ x = 4,\quad y = 10 \]

\[ \therefore \quad \text{The solution is } (4,\; 10). \]