Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{x}{7} + \frac{y}{3} = 5, \\ \frac{x}{2} – \frac{y}{9} = 6 \]

Solution

Step 1: Remove Fractions

Multiply the first equation by 21:

\[ 3x + 7y = 105 \quad \text{(1)} \]

Multiply the second equation by 18:

\[ 9x – 2y = 108 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 3x + 7y = 105 \]

\[ 7y = 105 – 3x \]

\[ y = \frac{105 – 3x}{7} \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 9x – 2\left(\frac{105 – 3x}{7}\right) = 108 \]

Multiply both sides by 7:

\[ 63x – 210 + 6x = 756 \]

\[ 69x = 966 \]

\[ x = 14 \]

Step 4: Find the Value of y

Substitute \(x = 14\) into equation (3):

\[ y = \frac{105 – 3(14)}{7} \]

\[ y = \frac{105 – 42}{7} = \frac{63}{7} = 9 \]

Conclusion

The solution of the given system of equations is:

\[ x = 14,\quad y = 9 \]

\[ \therefore \quad \text{The solution is } (14,\; 9). \]