Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ x + \frac{y}{2} = 4, \\ \frac{x}{3} + 2y = 5 \]

Solution

Step 1: Remove Fractions

Multiply the first equation by 2:

\[ 2x + y = 8 \quad \text{(1)} \]

Multiply the second equation by 3:

\[ x + 6y = 15 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 2x + y = 8 \]

\[ y = 8 – 2x \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ x + 6(8 – 2x) = 15 \]

\[ x + 48 – 12x = 15 \]

\[ -11x = -33 \]

\[ x = 3 \]

Step 4: Find the Value of y

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

\[ y = 8 – 2(3) \]

\[ y = 2 \]

Conclusion

The solution of the given system of equations is:

\[ x = 3,\quad y = 2 \]

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