The equations: x + 4y - 3 = 0 and 2x + 8y - 6=0 have

The correct answer is Option (4) → infinite solutions

Two linear equations \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\) have infinitely many solutions when \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\): the second equation is then only a multiple of the first, so both stand for the same line.

The three ratios:

• \(\displaystyle \frac{a_1}{a_2}=\frac{1}{2}\)

• \(\displaystyle \frac{b_1}{b_2}=\frac{4}{8}=\frac{1}{2}\)

• \(\displaystyle \frac{c_1}{c_2}=\frac{-3}{-6}=\frac{1}{2}\)

The same line (all three ratios are equal):

• \(2x+8y-6=0\) is twice the first equation

• So every point of the line \(x+4y=3\) satisfies both