Use elimination method to find all possible solutions of the following pair of linear equations : 2x + 3y = 8 (1) 4x + 6y = 7 (2)
Step-by-Step Solution
Key Concept: The elimination method involves adding or subtracting the equations after suitable multiplication so that one variable gets eliminated. If the resulting equations give a contradiction, the system has no solution (inconsistent). If they give an identity, the system has infinitely many solutions (dependent).
1. Write the given equations
$$\begin{aligned}
(1)\;& 2x + 3y = 8 \\
(2)\;& 4x + 6y = 7
\end{aligned}$$
2. Make the coefficients of one variable equal
Multiply equation (1) by 2 so that the coefficient of \(x\) (or \(y\)) matches that in equation (2):
$$2\times(1):\; 4x + 6y = 16$$
3. Subtract the two equations
$$\begin{aligned}
(4x + 6y = 16) \; - \; (4x + 6y = 7) \;\Rightarrow\; 0 = 9
\end{aligned}$$
4. Interpret the result
The statement \(0 = 9\) is a contradiction. Hence the two equations are inconsistent; they represent two parallel lines that never intersect.
5. Conclusion
Since the system is inconsistent, there is no solution (i.e., the pair of equations has no common point).
Correct Answer: No solution (the system is inconsistent).