Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ 21x + 47y = 110 \quad (1) \]

\[ 47x + 21y = 162 \quad (2) \]

Solution

Step 1: Express One Variable in Terms of the Other

From equation (1):

\[ 21x = 110 – 47y \]

\[ x = \frac{110 – 47y}{21} \quad (3) \]

Step 2: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 47\left(\frac{110 – 47y}{21}\right) + 21y = 162 \]

Multiply both sides by 21:

\[ 47(110 – 47y) + 441y = 3402 \]

\[ 5170 – 2209y + 441y = 3402 \]

\[ 5170 – 1768y = 3402 \]

\[ 1768y = 1768 \]

\[ y = 1 \]

Step 3: Find the Value of x

Substitute \(y = 1\) into equation (3):

\[ x = \frac{110 – 47}{21} \]

\[ x = \frac{63}{21} = 3 \]

Conclusion

The solution of the given system of equations is:

\[ x = 3,\quad y = 1 \]

\[ \therefore \quad \text{The solution is } (3,\; 1). \]