# linear equation with two variables. but topic says it is one question.

Here's the question: 4 chairs and three tables cost rs 2100 and 5 chairs and 2 tables cost rs. 1750 the cost of a table and a chair Separately are: There are some options . Please help me

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In some solutions, I saw cross multiplication. Also explain it. –  Govind Balaji Jan 24 at 17:09
Write down the equations: Let $C$ be the cost of a chair and $T$ the cost of a table. Then the first condition gives $4C+3T=1200$. What equation does the second condition give? Once you have the two equations, you need to solve them. Could you do this? –  David Mitra Jan 24 at 17:10

Let $c$ be the cost of a chair. Let $t$ be the cost of a table.

This gives you two equations in two variables:

$$4c + 3t = 2100\tag{1}$$ $$5c + 2t = 1750\tag{2}$$

Can you take it from here?

If you solve for, say, $c$ using the first equation: $$4 c = 2100 - 3t \iff c = \dfrac {2100}{4} - \dfrac {3t}{4} \iff 525 -\dfrac 34 t$$ You can then substitute this $c$ into the second equation, which will then be an equation in one variable, $t$: $$5\;\left(\underbrace{525 - \frac 34 t}_{\large = \;c}\right) + 2t = 1750$$ Solve for $t$, and then compute $c = 525-\dfrac 34t$.

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I can take till here, but not further any more –  Govind Balaji Jan 24 at 17:12
I added a bit more in the way of how to proceed. Work on it a bit more; check back with me if you stay stuck. But put some thought into this, first. –  amWhy Jan 24 at 17:14
Expand $$5\left(525 - \frac 34 t\right) + 2t = 1750$$ $$2625 - \frac{15}{4}t + 2t = 1750\iff -\frac 74 t= -875 \iff t = \frac 47 \cdot 875\iff t = 500$$ Now using $t = 500$, solve $c = 525 - \frac 34t$ –  amWhy Jan 24 at 17:26

1 - Call cost of chair of $C$ 2 - Call cost of tables of $T$ as @David Mitra said.

$\begin{cases} 4C + 3T = 2100\\ 5C + 2T = 1750 \end{cases}$

You need to separate one those variables, example $T$:

$3T = 2100 - 4C\Rightarrow T = 700 - \frac{4C}{3}$

Finally, second equation: $5C + 2(700-\frac{4C}{3}) = 1750$

Does that help?

Continue:

$5C + 1400 - \dfrac83 C = 1750\Rightarrow \dfrac 73C = 350\Rightarrow \boxed{C = 150}$

So,

$T = 700 - \dfrac 43 .150 = 700 - 4.50 = 700 - 200 \Rightarrow \boxed{T = 500}$