# How to find the point of intersection of two lines?

How to find the point of intersection of two lines ?

Let's say $2x+y=5$ and $3x+2y=5$ ?

Thanks

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Have you encountered systems of equations until now? Finding the intersection of two lines means solving the system of equations determined by those two lines. – Beni Bogosel Jun 29 '11 at 7:44

The graphs of two lines intersect at a point $(s,t)$ when the point $(s,t)$ is on both lines.

The point $(s,t)$ lies on the line $ax+by=c$ when $as+bt=c$.

Thus, to find the intersection of two lines $ax+by=c$ and $dx+ey=f$ (if it exists), we want to look for a pair of numbers $s$ and $t$ for which $$as+bt=c\qquad\text{and}\qquad ds+et=f.$$

In your example, we want to find an $s$ and $t$ for which $2s+t=5$ and $3s+2t=5$.

Note that if $2s+t=5$, then $t=5-2s$. Substitute this in for $t$ in the expression $3s+2t=5$ to find the value for $s$. Then solve for $t$ using the fact that $t=5-2s$.

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ahah i didnt knew that the point of intersection is the same as the point we get when we solve it. Anyways thanks :) :) – kritya Jun 29 '11 at 7:43
@kritya: No problem. The correspondence between the graph of a line (or any function), and its equation, is a very important one. – Zev Chonoles Jun 29 '11 at 7:46