Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations:

\[ 7(y+3) – 2(x+2) = 14, \\ 4(y-2) + 3(x-3) = 2 \]

Solution

Step 1: Simplify Both Equations

First equation:

\[ 7(y+3) – 2(x+2) = 14 \]

\[ 7y + 21 – 2x – 4 = 14 \]

\[ -2x + 7y + 17 = 14 \]

\[ -2x + 7y = -3 \quad \text{(1)} \]

Second equation:

\[ 4(y-2) + 3(x-3) = 2 \]

\[ 4y – 8 + 3x – 9 = 2 \]

\[ 3x + 4y – 17 = 2 \]

\[ 3x + 4y = 19 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (1):

\[ -2x + 7y = -3 \]

\[ 7y = 2x – 3 \]

\[ y = \frac{2x – 3}{7} \quad \text{(3)} \]

Step 3: Substitute in Equation (2)

Substitute equation (3) into equation (2):

\[ 3x + 4\left(\frac{2x – 3}{7}\right) = 19 \]

Multiply both sides by 7:

\[ 21x + 8x – 12 = 133 \]

\[ 29x = 145 \]

\[ x = 5 \]

Step 4: Find the Value of y

Substitute \(x = 5\) into equation (3):

\[ y = \frac{2(5) – 3}{7} \]

\[ y = \frac{10 – 3}{7} = 1 \]

Conclusion

The solution of the given system of equations is:

\[ x = 5,\quad y = 1 \]

\[ \therefore \quad \text{The solution is } (5,\; 1). \]