Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ 3x – \frac{y+7}{11} + 2 = 10, \\ 2y + \frac{x+11}{7} = 10 \]

Solution

Step 1: Simplify Both Equations

First equation:

\[ 3x – \frac{y+7}{11} + 2 = 10 \]

\[ 3x – \frac{y+7}{11} = 8 \]

Multiply both sides by 11:

\[ 33x – (y+7) = 88 \]

\[ 33x – y = 95 \quad \text{(1)} \]

Second equation:

\[ 2y + \frac{x+11}{7} = 10 \]

\[ 2y = 10 – \frac{x+11}{7} \]

Multiply both sides by 7:

\[ 14y = 70 – x – 11 \]

\[ 14y = 59 – x \]

\[ x = 59 – 14y \quad \text{(2)} \]

Step 2: Substitute in Equation (1)

Substitute equation (2) into equation (1):

\[ 33(59 – 14y) – y = 95 \]

\[ 1947 – 462y – y = 95 \]

\[ 1947 – 463y = 95 \]

\[ 463y = 1852 \]

\[ y = 4 \]

Step 3: Find the Value of x

Substitute \(y = 4\) into equation (2):

\[ x = 59 – 14(4) \]

\[ x = 59 – 56 = 3 \]

Conclusion

The solution of the given system of equations is:

\[ x = 3,\quad y = 4 \]

\[ \therefore \quad \text{The solution is } (3,\; 4). \]