Pair of Linear Equation in Two Variables- Mindspark

What is Pair Linear Equation in Two Variables?

A linear equation in two variables is written as ax + by + c = 0. Where a and b are coefficients of variables x and y respectively and c is a constant. Coefficient means the numbers which are with x and y, constant means that in a question their value cannot be changed and variable means that its value can be changed with the context of the question. In the given equation a,b,c are real numbers and a,b are the coefficients of x and y and a or b cannot be zero. A pair of linear equations in two variables means that in a question we have two equations in the form of ax + by + c = 0 and we have to find the possible solutions of the variables x and y. 

Linear Equation in Two Variables

The linear equation in two variables in the form of ax + by + c = 0 generally represents a line on the coordinate axes (graph). The easiest way to represent the equation on the graph is using the substitution method, i.e first assuming a value of a variable and finding out the value of the other.

Example:

3x + 2y – 6 = 0 

The above equation has two variables i.e, x and y.

Graphically we can represent this equation by substituting the variables to zero.

The value of x when y=0 is, we can write the equation as –

3x + 2(0) = 6

⇒ x = 2

We show this solution on the graph as (2,0)

Therefore the value of y when x = 0 is,

3(0) + 2y = 6

⇒ y = 3

We show this solution on the graph as (0, 3)

Types of Solution of Pair of Linear Equation with Two Variables-

Unique Solution – When the lines represented on the coordinate axes by the equations intersect each other, then we say that the pair of equations has a unique solution.

No Solution – When the lines represented on the coordinate axes by the equations do not intersect and are parallel to each other, then we say that the pair of equations has no solution.

Infinite Solutions – When the lines represented on the coordinate axes by the equations are coincident i.e, they both overlap each other ( form the same line) then we say that the pair of equations have infinite solutions.

Methods To Solve Pair Of Linear Equation In Two Variables-

1. Graphical Method :

• First, we plot the two-variable equations on the coordinate axes(graph).
• Then we see where both the lines are intersecting.
• The intersecting point is the solution.

Example: Find the value of x and y by solving the given equations by graphical method.
x – y + 4 = 0 – (1) and x + y – 10 = 0 – (2)

First, we solve each equation separately and plot it. While plotting it on the axes, we find a point of intersection between the two equations. The point of intersection will be the solution.

As we can see in the figure given below, the point of intersection is (3,7).

Therefore the solution is x = 3 and y = 7.

2. Substitution Method :

• First, solve both the equations for one variable.
• Then by using the first variable, find the value of the second variable.

Example : Solve the pair of equations by substitution method: 

5x – 8y = 2                           – (1)

 x + 4y = 3                            – (2)

We can  pick either of the equations and write one variable in terms of the other, here taking equation (2) first:

So, we can write  x + 4y = 3 as    x = 3 – 4y    – (3)

After substituting the value of x in equation (1), we get –

5 (3 – 4y) + 8y = 2

⇒ 15 – 20y + 8 = 2

⇒ 23 – 20y = 2

⇒ – 20y = -21

⇒ y = 21/20

Now we substitute the value of y in equation (3),

x = 3 – 4 (21/20)

⇒ x = 3 – 21/5

⇒ x = (15-21)/5

⇒ x = – 6/5

Therefore the solution is x = -6/5 and y = 21/20.

3. Cross Multiplication Method :

• Consider pair of linear equation in two variables as a_1\text{x} + b_1\text{y} +c_1=0 \text{ and } a_2\text{x}=b_2\text{y} +c_2.
• Now write the variables, coefficients and the constant in the following way-

•  Now we write the equation by cross multiplying and subtracting the product.

•   Now after solving the equations for x and y we get : 

Example : Solve the following pair of linear equation by cross multiplication method 

3x – 5y = – 2  and y – 2x = 6

The given equations can be written as: 

3x – 5y +2  = 0     – (1)

-2x + y – 6 = 0     – (2)

Now by using cross multiplication we get 

x=\frac{b_{1} \cdot c_{2}-b_{2} \cdot c_{1}}{a_{1} \cdot b_{2}-a_{2} \cdot b_{1}}

\Rightarrow x=\frac{(-5) \times(-6)-(1) \times(2)}{(3) \times(1)-(-2) \times(-5)}

\Rightarrow x=\frac{30-2}{3-10}

\Rightarrow x=\frac{28}{-7}

\Rightarrow x=-4

\mathrm{y}=\frac{c_{1} \cdot a_{2}-c_{2} \cdot a_{1}}{a_{1} \cdot b_{2}-a_{2} \cdot b_{1}}

\Rightarrow y=\frac{(2) \times(-2)-(-6) \times(3)}{(3) \times(1)-(-2) \times(-5)}

\Rightarrow y=\frac{(-4)-(-18)}{3-10}

\Rightarrow y=\frac{14}{-7}

\Rightarrow y=-2

Therefore the solution is x = -4 and y = -2.