# Intersection of two lines using general form

How do I find the intersection of these two lines with their equations in general form. I don't want to graph them and I'm wondering if its possible with out converting them to gradient intercept form?

$2x+3y-5=0$ and $5x-y-4=0$

Thank you!

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From the second equation, we have that $y=5x-4$. Plugging that into the first equation we get $$0 = 2x+3y - 5 = 2x + 3(5x-4) -5 = 2x + 15x - 12 - 5 = 17x-17.$$ Solving for $x$, and then plugging into $y=5x-4$, gives the point of intersection.

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@Eugene: No, you're right. – Arturo Magidin Jun 10 '12 at 1:27

We will label the two equations: $$2x +3y = 5 \qquad (1)$$ and $$5x - y = 4 \qquad (2)$$

Multiply the second equation by $3$ to get $$15 x -3y = 12 \qquad (3)$$

By adding $(3)$ to $(1)$ we get $$17x = 17.$$ Therefore $x = 1$. Now plugging in $x = 1$ to $(2)$ we get $$5 -y = 4$$ which means $$- y = -1$$ and so $y = 1$.

We check and find that it is indeed the case ie $$2(1) + 3(1) = 5$$ and $$5(1) - 1 = 4.$$

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