What are the values of x and y?

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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

$7 (3 - 2 y) - 15 y = 2$

$21 - 14 y - 15 y = 2$

$- 29 y = 2 - 21$

$- 29 y = - 19$

$y = 19 / 29$

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

$2 x + 3 y = 8$ ( i )

$4 x + 5 y = 7$ ( 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:

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

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

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

$y = 9$