Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

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

Solution

Step 1: Express One Variable in Terms of the Other

From the first equation:

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

\[ x = \frac{3}{2} – 2y \quad \text{(1)} \]

Step 2: Substitute in the Second Equation

Substitute equation (1) into the second equation:

\[ 2\left(\frac{3}{2} – 2y\right) + y = \frac{3}{2} \]

\[ 3 – 4y + y = \frac{3}{2} \]

\[ 3 – 3y = \frac{3}{2} \]

\[ 3y = \frac{3}{2} \]

\[ y = \frac{1}{2} \]

Step 3: Find the Value of x

Substitute \(y = \frac{1}{2}\) into equation (1):

\[ x = \frac{3}{2} – 2\left(\frac{1}{2}\right) \]

\[ x = \frac{3}{2} – 1 = \frac{1}{2} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{1}{2},\quad y = \frac{1}{2} \]

\[ \therefore \quad \text{The solution is } \left(\frac{1}{2},\; \frac{1}{2}\right). \]