# Linear Equation in two variables problem

37 Pens and 53 pencils together cost Rs. 320 while 53 Pens and 37 Pencils together cost Rs. 400, Find the cost of a Pen and that of a Pencil.

So far I had done the following: Let cost of 1 Pen be Rs.x

And let cost of 1 Pencil be Rs.y

So, equations will be:

37 x + 53 y = 320   -----------  (1)

53 x + 37 y = 400   -----------  (2)

Now which formula I should apply to solve this linear equation in two variables.

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should likely tag as linear algebra and homework. What have you tried so far? –  gt6989b Sep 19 '12 at 16:05

Let $x$ be the cost of a pen, and let $y$ be the cost of a pencil.

Then $37x+53y=320$ and $53x+37y=400$. We have two linear equations in two unknowns. In principle this system of equations is routine to solve for $x$ and $y$, but it might be kind of messy.

But note the nice partial symmetry, and observe that $$(37x+53y)+(53x+37y)=90x+90y=90(x+y).$$

Remark: So now we know that the combined cost of a pen and pencil is $8$. So we are finished. But what about the individual costs? Note that $(53x+37y)-(37x+53y)=16(x-y)=400-320$. So $x-y=5$. Now we can easily find $x$ and $y$. For $(x+y)+(x-y)=2x=13$ and therefore $x=6.5$. It follows that $y=1.5$.

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