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.
Write both in standard form.
\(-x + 2y + 2 = 0\)
\(\dfrac{1}{2}x - \dfrac{1}{4}y + 1 = 0\)
Compare ratios of coefficients.
\(\dfrac{a_1}{a_2} = \dfrac{-1}{\,1/2\,} = -2\)
\(\dfrac{b_1}{b_2} = \dfrac{2}{\,-1/4\,} = -8\)
Since these ratios are unequal, the lines intersect at exactly one point.
Therefore, there is a unique solution.