All about Linear Equation in Two Variables and pair of Linear Equation in Two Variables

 

Linear Equation in Two Variables 

pair of Linear Equation in Two Variables

 

What is Linear Equation in Two Variables?

Linear equation is a mathematical equation which can be represented in the form of the equation ax+ by+ c = 0 is called linear equation in two variables.

Where a, b, c are real number.  Both a and b are not zero at a time.

 

Example:

"Sum of two numbers is 100" we can represent the statement in the form of mathematical equation as x + y = 100 (so, there two variables are x and y). There are infinitely many values of x and correspondingly of y.

 

"Difference of 3 times of A's tuition fee and 2 times of B's tuition fee is Rs.50. For this statement the mathematical equation can be 3A-2B=50

 

Types of Question with Answer:

1.     Express the linear equation 7 = 2x in the form of ax + by + c = 0 and also write the values of a, b, and c.

2.     The cost of a toy telephone is the same as the cost of 3 balls. Express the statement in the form of linear equation in two variables

 

Solution of Linear Equation:

The values of the two variables that satisfy the equation are the solutions of the given linear equation.

 

X+2Y = 10 (If we put x=4 and y=3)

LHS 4 + 2 x 3 = 4 + 6 = 10= RHS

 

(If we put x= 2 and y = 4)

LHS 2 + 2 x 4= 2 + 8 = 10 = 10 RHS

Therefore a linear equation in two variables can have infinitely many solutions.

 

Types of Questions with Answer

 

Find the value of k so that x=1, y= -1 is a solution of 2x +KY = 19. Find two more solutions of resulting equation.

 

Graph of a linear Equation in two variables

2x + 5y = 12, Three solutions for this equation may be (1, 2), (-4, 4) and (6, 0).We can plot these points as A, B and C on graph paper.

 

Pair of Linear Equation in Two Variables

 

We may present a pair of linear equations in two variables as follow a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0.

When we draw the graphs of these two linear equations on the same graph paper, we can have two lines which are

1.     Intersecting at a point.

2.     Coincident to each other.

3.     Parallel to each other.

 

Graphical Method of solution of a pair of linear equations

 

7x-y=42----1

 

x	6	5
y	0	-7

 

3x-y=-6----2

 

x	0	-2
y	6	0

 

Outcome:

1.     When two lines are intersecting the solution will be unique. It means there is only one common solution of both the equations.

2.     If two lines are coinciding there are infinitely many solutions of the pair of equations.

3.     When two lines are parallel to each other then there is no solution of the pair of equations.

 

Therefore when the two lines are either intersecting or coincident we say that the given linear equations are consistent. If both are parallel we say them inconsistent.

Coefficient comparison method to know consistent or inconsistent of two linear equations

 

a1/a2≠b1/b2          Unique Solution   Intersecting lines   Consistent

a1/a2=b1/b2=c1/c2     Infinitely many Solution Coinciding lines Consistent & Dependent

a1/a2=b1/b2≠c1/c2         No solution Parallel lines          Inconsistent

 

Example :

3x+2y=5, 2x-3y=7

a1=3, b1=2, c1=-5

a2=2, b2=-3, c2=-7

a1/a2=3/2, b1/b2=2/(−3), c1/c2=(−5)/(−7)

 

a1/a2≠b1/b2   because 3/2≠2/(−3)

 so the linear pair is consistent.

 

Algebraic method of solving a pair of linear equations

 

Substitution Method

 

In this method we take any one of the given pair of linear equations and express one variable in terms of the other variable,

 

Example:

2x+3y=7  ---(1)

3x+5y=18 ---(2)

 

From (1) y=(7−2x)/3

 

Put the value of Y in (2)

3x+5 ((7−2x)/3)=18

⟹3 x 3x+35 −10x=18 x 3

⟹9x+35 −10x=54

⟹−x=54−35=19

X=-19

Put the value of x in (1)

Y=(7−2 x (−19))/3=(7+38)/3=45/3=15

 

The solution is x= -19, y= 15

 

 

Put the value of x in (1)

Y = (8−2)/3=6/3=2

∴x=1, y=2