Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ \frac{x}{3} + \frac{y}{4} = 11, \\ \frac{5x}{6} – \frac{y}{3} = -7 \]

Solution

Step 1: Remove Fractions

Multiply the first equation by 12:

\[ 4x + 3y = 132 \quad \text{(1)} \]

Multiply the second equation by 6:

\[ 5x – 2y = -42 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ 4x + 3y = 132 \]

\[ 3y = 132 – 4x \]

\[ y = \frac{132 – 4x}{3} \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 5x – 2\left(\frac{132 – 4x}{3}\right) = -42 \]

Multiply both sides by 3:

\[ 15x – 264 + 8x = -126 \]

\[ 23x = 138 \]

\[ x = 6 \]

Step 4: Find the Value of y

Substitute \(x = 6\) into equation (3):

\[ y = \frac{132 – 4(6)}{3} \]

\[ y = \frac{132 – 24}{3} = \frac{108}{3} = 36 \]

Conclusion

The solution of the given system of equations is:

\[ x = 6,\quad y = 36 \]

\[ \therefore \quad \text{The solution is } (6,\; 36). \]