Question

If3x+5y=9and5x+3y=7then,what is the value of x+y=​

Answer 1

3x + 5y = 9 and 5x + 3y = 7. What is the value of x+y?

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These is a problem of linear equations.

The problems of can be solved by two different methods…

Elemination

Substitution

Here we will discuss about the elemination techniques.

As the name suggest that we need to eleminate the variables (either x or y) to get the answer. Let's understand with the help of an example.

Given eqn

3x+5y=9

5x+3y=7

In order to eleminate “x”we need to make the co-efficient of “x” same on both the equation.

To achieve this

we will multiply 5 with equation 1

And multiply 3 with equation 2

On multiplication it will look like….

15x+25y=45

15x+9y=21

Now we will substract equation 1 and 2, it will give,

16y=24

y=1.5

Now, we will put y = 1.5 in equation 1 and solve

Upon solving we get,

x=0.5

Now, we got both x and y.

So, x+y=0.5+1.5=2

Answer 2

Given:-

3x + 5y = 9

5x + 3y = 7

To find:-

Value of x+y

Method:-

Substitution Method

Solution:-

$\sf{3x + 5y = 9 \longrightarrow (i)}$

$\sf{5x + 3y = 7 \longrightarrow (ii)}$

From eq.(i)

$\sf{3x + 5y = 9}$

= $\sf{3x = 9 - 5y}$

$\sf{\implies x = \dfrac{9-5y}{3}}$

Substituting the value of x in eq.(ii)

$\sf{5x + 3y = 7}$

= $\sf{5\times\bigg( \dfrac{9-5y}{3}\bigg) + 3y = 7}$

$\sf{\implies \dfrac{45 - 25y}{3} + 3y = 7}$

$\sf{\implies \dfrac{45 - 25y + 9y}{3} = 7}$

$\sf{\implies 45 - 16y = 7\times 3}$

$\sf{\implies 45-16y = 21}$

$\sf{\implies -16y = 21-45}$

$\sf{\implies -16y = -24}$

$\sf{\implies y = \dfrac{-24}{-16}}$

$\sf{\implies y = \dfrac{3}{2}}$

Putting the value of y in eq.(i)

$\sf{3x + 5y = 9}$

= $\sf{3x + 5\times\dfrac{3}{2} = 9}$

$\sf{\implies 3x + \dfrac{15}{2} = 9}$

$\sf{\implies \dfrac{6x + 15}{2} = 9}$

$\sf{\implies 6x + 15 = 9\times2}$

$\sf{\implies 6x + 15 = 18}$

$\sf{\implies 6x = 18 - 15}$

$\sf{\implies 6x = 3}$

$\sf{\implies x = \dfrac{3}{6}}$

$\sf{\implies x = \dfrac{1}{2}}$

Now,

$\sf{x+y = \dfrac{3}{2} + \dfrac{1}{2}}$

= $\sf{ \dfrac{3+1}{2}}$

= $\sf{\dfrac{4}{2}}$

= $\sf{2}$

$\sf{\therefore The\:value\:of\:x+y\:is\:2}$