Do the following pair of linear equations have no solution? Justify your answer.
(i) \(2x + 4y = 3\), \(12y + 6x = 6\)
(ii) \(x = 2y\), \(y = 2x\)
(iii) \(3x + y - 3 = 0\), \(\dfrac{2}{3}x + \dfrac{1}{2}y = 2\)
(i) Yes, (ii) No, (iii) No
Do the following equations represent a pair of coincident lines? Justify your answer.
(i) \(3x + \dfrac{1}{7}y = 3\), \(7x + 3y = 7\)
(ii) \(-2x - 3y = 1\), \(6y + 4x = -2\)
(iii) \(\dfrac{x}{2} + \dfrac{y}{5} + \dfrac{5}{16} = 0\), \(4x + 8y + \dfrac{5}{4} = 0\)
(i) No, (ii) Yes, (iii) No
Are the following pair of linear equations consistent? Justify your answer.
(i) \(-3x - 4y = 12\), \(4y + 3x = 12\)
(ii) \(\dfrac{3}{5}x - y = 12\), \(\dfrac{1}{5}x - 3y = 16\)
(iii) \(2ax + by = a\), \(4ax + 2by - 2a = 0\); \(a,b \ne 0\)
(iv) \(x + 3y = 11\), \(2(2x + 6y) = 22\)
(i) Inconsistent, (ii) Consistent, (iii) Consistent, (iv) Inconsistent
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.
For all real values of \(c\), the pair of equations \(x - 2y = 8\) and \(5x - 10y = c\) have a unique solution. Justify whether it is true or false.
False.
The line represented by \(x = 7\) is parallel to the x–axis. Justify whether the statement is true or not.
False. \(x = 7\) is a vertical line, hence parallel to the y–axis.
Sample Question 1: Is it true to say that the pair of equations \(-x + 2y + 2 = 0\) and \(\dfrac{1}{2}x - \dfrac{1}{4}y = -1\) has a unique solution? Justify your answer.
Yes.
Sample Question 2: Do the equations \(4x + 3y - 1 = 5\) and \(12x + 9y = 15\) represent a pair of coincident lines? Justify your answer.
No.
Sample Question 3: Is the pair of equations \(x + 2y - 3 = 0\) and \(3x + 6y - 9 = 0\) consistent? Justify your answer.
Yes. They are dependent (coincident) and hence consistent.