Solve the System of Equations by the Substitution Method

Video Explanation

Question

Solve the following system of equations by the substitution method:

\[ 11x + 15y + 23 = 0 \\ , 7x – 2y – 20 = 0 \]

Solution

Step 1: Write the Equations in Standard Form

\[ 11x + 15y = -23 \quad \text{(1)} \]

\[ 7x – 2y = 20 \quad \text{(2)} \]

Step 2: Express One Variable in Terms of the Other

From equation (2):

\[ 7x – 2y = 20 \]

\[ 2y = 7x – 20 \]

\[ y = \frac{7x – 20}{2} \quad \text{(3)} \]

Step 3: Substitute in Equation (1)

Substitute equation (3) into equation (1):

\[ 11x + 15\left(\frac{7x – 20}{2}\right) = -23 \]

Multiply both sides by 2:

\[ 22x + 105x – 300 = -46 \]

\[ 127x = 254 \]

\[ x = 2 \]

Step 4: Find the Value of y

Substitute \(x = 2\) into equation (3):

\[ y = \frac{7(2) – 20}{2} \]

\[ y = \frac{14 – 20}{2} = -3 \]

Conclusion

The solution of the given system of equations is:

\[ x = 2,\quad y = -3 \]

\[ \therefore \quad \text{The solution is } (2,\; -3). \]