Solve the System of Equations by the Substitution Method
Video Explanation
Question
Solve the following system of equations:
\[ 23x – 29y = 98, \\ 29x – 23y = 110 \]
Solution
Step 1: Express One Variable in Terms of the Other
Subtract the first equation from the second:
\[ (29x – 23y) – (23x – 29y) = 110 – 98 \]
\[ 6x + 6y = 12 \]
\[ x + y = 2 \quad \text{(1)} \]
\[ y = 2 – x \quad \text{(2)} \]
Step 2: Substitute in the First Equation
Substitute equation (2) into \(23x – 29y = 98\):
\[ 23x – 29(2 – x) = 98 \]
\[ 23x – 58 + 29x = 98 \]
\[ 52x = 156 \]
\[ x = 3 \]
Step 3: Find the Value of y
Substitute \(x = 3\) into equation (2):
\[ y = 2 – 3 = -1 \]
Conclusion
The solution of the given system of equations is:
\[ x = 3,\quad y = -1 \]
\[ \therefore \quad \text{The solution is } (3,\; -1). \]