Which of the following pair of linear equations has / have infinitely many solutions?
A. \(9x+3y+12=0\); and \(18x+6y+24=0\)
B. \(2x-3y=13\); and \(7x-2y=20\)
C. \(x+y=5\); and \(2x+2y=10\)
D. \(3x+y-3=0\); and \(2x+\frac{2}{3}y=2\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A, C and D only
A pair of linear equations \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\) has infinitely many solutions when \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\): one equation is a multiple of the other, so both describe the same line.
A. \(9x+3y+12=0\); and \(18x+6y+24=0\) Correct. All three ratios are \(\frac{1}{2}\) (\(\frac{9}{18}\), \(\frac{3}{6}\), \(\frac{12}{24}\)); the second equation is twice the first.
B. \(2x-3y=13\); and \(7x-2y=20\) Incorrect. \(\frac{a_1}{a_2}=\frac{2}{7}\) but \(\frac{b_1}{b_2}=\frac{3}{2}\), so the lines cross at one point and there is only one solution.
C. \(x+y=5\); and \(2x+2y=10\) Correct. All three ratios are \(\frac{1}{2}\) (\(\frac{1}{2}\), \(\frac{1}{2}\), \(\frac{5}{10}\)); the second equation is twice the first.
D. \(3x+y-3=0\); and \(2x+\frac{2}{3}y=2\) Correct. All three ratios are \(\frac{3}{2}\) (\(\frac{3}{2}\), \(\frac{1}{2/3}\), \(\frac{3}{2}\)); the first equation is \(\frac{3}{2}\) times the second.