# 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 $\mathrm{Rs}.x$

And let cost of $1$ Pencil be $\mathrm{Rs.}y$

So, equations will be:

$37 \cdot x + 53\cdot y = 320 \mathrm{-----} (1)$

$53 \cdot x + 37 \cdot y = 400 \mathrm{-----} (2)$

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

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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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x - cost of a pen y - cost of a pencil

37x + 53y = 320

53x + 37y = 400

you should be able to proceed. Try find value of x from one equality an set it up to second. then you have equality with y only, so can find an answer

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