For the pair of equations \(\lambda x + 3y = -7\) and \(2x + 6y = 14\) to have infinitely many solutions, the value of \(\lambda\) should be 1. Is the statement true? Give reasons.
No.
Idea. For two equations to have infinitely many solutions, the lines must be coincident. This happens when all three ratios are equal:
\(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\).
Step 1: Write in standard form.
First equation: \(\lambda x + 3y = -7\)
Move terms: \(\lambda x + 3y + 7 = 0\)
So \(a_1 = \lambda\), \(b_1 = 3\), \(c_1 = 7\).
Second equation: \(2x + 6y = 14\)
Move terms: \(2x + 6y - 14 = 0\)
So \(a_2 = 2\), \(b_2 = 6\), \(c_2 = -14\).
Step 2: Form the ratios.
\(\dfrac{a_1}{a_2} = \dfrac{\lambda}{2}\)
\(\dfrac{b_1}{b_2} = \dfrac{3}{6} = \dfrac{1}{2}\)
\(\dfrac{c_1}{c_2} = \dfrac{7}{-14} = -\dfrac{1}{2}\)
Step 3: Compare the ratios.
The second ratio is \(\dfrac{1}{2}\), while the third is \(-\dfrac{1}{2}\).
Since one is positive and the other is negative, they cannot be equal, no matter what value of \(\lambda\) is chosen.
Conclusion. The statement is false. There is no value of \(\lambda\) for which the pair of equations has infinitely many solutions.