Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

Let $a, b$ , and $c$ be real numbers. Consider the system of simultaneous equations in variables $x$ and $y:$ $a x+b y =c-1$ and $(a+5) x+(b+3) y =c+1$ . Determine the value(s) of $c$ in terms of $a$ such that the system always has a solution for any $a$ and $b$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We have to only consider when the determinant of $\begin{pmatrix}a & b \\ a+5 & b+3\end{pmatrix}$ is zero. That is, when $b=3 a / 5$ . Plugging in $b=3 a / 5$ , we find that $(a+5)(c-1)=a(c+1)$ or that $c=2 a / 5+1$ .

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