Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.9 AIME Prove it United States

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

Solution

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 $b = \dfrac{3a}{5}$ . Plugging in $b = \dfrac{3a}{5}$ , we find that $(a+5)(c-1) = a(c+1)$ or that $c = \dfrac{2a}{5} + 1$ .

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