Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.5 Junior Find the answer

Given lines $l\_1$ : $ax + y + 3 = 0$ and $l\_2$ : $x + (2a - 3)y = 4$ , where $l\_1 \perp l\_2$ , find the value of $a$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Since lines $l\_1$ : $ax + y + 3 = 0$ and $l\_2$ : $x + (2a - 3)y = 4$ are perpendicular, the sum of the products of their respective $x$ and $y$ coefficients is equal to zero. Therefore, we have:

$a + (2a - 3) = 0$

Solving this equation:

$3a - 3 = 0$

$3a = 3$

$a = \boxed{1}$

Thus, the answer is $1$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.