Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{x}{2} + y = 0.8, \\ \frac{7}{x + \frac{y}{2}} = 10 \]

Solution

Step 1: Remove Fractions

From the first equation:

\[ \frac{x}{2} + y = 0.8 \]

Multiply both sides by 10:

\[ 5x + 10y = 8 \quad \text{(1)} \]

From the second equation:

\[ \frac{7}{x + \frac{y}{2}} = 10 \]

Multiply both sides by \(x + \frac{y}{2}\):

\[ 7 = 10\left(x + \frac{y}{2}\right) \]

Multiply both sides by 2:

\[ 14 = 20x + 10y \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 5x + 10y = 8 \]

\[ 10y = 8 – 5x \]

\[ y = \frac{8 – 5x}{10} \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 14 = 20x + 10\left(\frac{8 – 5x}{10}\right) \]

\[ 14 = 20x + 8 – 5x \]

\[ 14 = 15x + 8 \]

\[ 15x = 6 \]

\[ x = \frac{2}{5} \]

Step 4: Find the Value of y

Substitute \(x = \frac{2}{5}\) into equation (3):

\[ y = \frac{8 – 5\left(\frac{2}{5}\right)}{10} \]

\[ y = \frac{8 – 2}{10} \]

\[ y = \frac{6}{10} = \frac{3}{5} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{2}{5},\quad y = \frac{3}{5} \]

\[ \therefore \quad \text{The solution is } \left(\frac{2}{5},\; \frac{3}{5}\right). \]