Solve the System of Equations by the Substitution Method
Video Explanation
Question
Solve the following system of equations:
\[ 152x – 378y = -74, \\ -378x + 158y = -604 \]
Solution
Step 1: Express One Variable in Terms of the Other
From the first equation:
\[ 152x – 378y = -74 \]
\[ 152x = 378y – 74 \]
\[ x = \frac{378y – 74}{152} \quad \text{(1)} \]
Step 2: Substitute in the Second Equation
Substitute equation (1) into \(-378x + 158y = -604\):
\[ -378\left(\frac{378y – 74}{152}\right) + 158y = -604 \]
Multiply both sides by 152:
\[ -378(378y – 74) + 24016y = -91808 \]
\[ -142884y + 27972 + 24016y = -91808 \]
\[ -118868y = -119780 \]
\[ y = 1 \]
Step 3: Find the Value of x
Substitute \(y = 1\) into equation (1):
\[ x = \frac{378(1) – 74}{152} \]
\[ x = \frac{304}{152} = 2 \]
Conclusion
The solution of the given system of equations is:
\[ x = 2,\quad y = 1 \]
\[ \therefore \quad \text{The solution is } (2,\; 1). \]