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# What are the values of x and y?

The question is indented to solve the equation having two variables x and y.

In algebra, an equation can be defined as a mathematical statement which contains an “=” symbol between two algebraic expressions that have same value.

The most basic and common algebraic equations consist of one or more variables.

An equation having two variables is called a quadratic equation.

Let’s discuss how to solve the equation in two variables say x and y.

The methods to find the values of x and y are

### 1). Substitution Method

To understand this method, it’s important to know the steps:

Step 1 : First of all find the value of one variable , say  y in terms of other variable  that is x from either equation, whichever is convenient

Step 2: Substitute this value of y in other equation and reduce it to an equation in one variable that is in terms of x which can be solved and we will get the value of x.

Step 3: Now substitute value of x in the equation used in Step 1 to obtain the value of other variable.

Take an example to clear this method.

Example:

Solve the following pair of equations by substitution method:

$\displaystyle ~7x~~-~~15y~~~=~~2$                  (1)

$\displaystyle ~x~~+~~2y~~~=~~3$                  (2)

Step 1:

Let find the value of x from equation (2) and we get

$\displaystyle ~x~~+~~2y~~~=~~3$

$\displaystyle ~x~ = ~~3~~~-~~2y$                  (3)

Step 2 :

Substitute the value of x in equation   (1) we get

7  ( 3 – 2y ) − 15y  =2

21  −  14y   −  15y  =  2

−29y  = 2 – 21

−29y  =  −19

y  =  19 /  29

Step 3 :

Substitute value of y in equation (3), we get

x = 3 − 2 (19 / 29)

x = 49 / 29

Therefore the  solution is

x = 49 / 29 and  y =  19/ 29

### 2). Elimination Method

Steps to follow in this method are:

Step 1:   First step is to make the coefficient of one variable either x or y same in both the equations.

Step 2: Then add or subtract one equation from the other so that the same variable gets eliminated. Now, we will get the equation in one variable.

Step 3: Solve the equation in one variable and find out the value.

Step 4: Substitute this value of x or y in either of the original equations to get the value of the other variable.

See the example below to clear this method.

Example :

Solve the following equations by Substitution Method

2x  +  3y  =  8                       ( i )

4x  +  5y  =  7                        ( ii )

Sol: Multiply equation (i) by 2 and equation (ii) by 1 to make the coefficients of x equal. Then we get the equations as:

4x  +  6y  =  16                     (iii)   [on multiplying equation (i) by 2]

4x  +  5y  =  7                      (iv)   [on multiplying equation (ii) by 1]

Subtracting equation (iv) from equation (iii), we get

y   =   9

Put this value of y in either equation (i) or   equation (ii)

We put value of y in equation (i)

2x  +  3 ( 9 )  =  8

2x  +  27    =  8

2x  =  8  −  27

2x   =  − 19

x  =  − 19 / 2

So , the solution is

x  =  − 19 / 2   ,   y   =  9

These are the two methods to solve the equation for   x and   y.