Pair of Linear Equations in Two Variables

Pair of Linear Equations in Two Variables
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A linear equation in two variables x & y can be put in the form, ax + by + c = 0, where a,b & c are real numbers and a & b are not both zero (we often denote the condition a & b are not both zero by a2 + b2 ≠ 0)

Solution of such an equation is pair of values one for x & the other for (an ordered pair of x & y).

e.g.,

x + y =7

If we put x = 0 then y = 7

If we put y = 0 then x = 7

If we put x = 3 then y = 4

Then, (0,7), (7,0), (3,4) are solutions of y =7, we can find many in the similar way.

A linear equation in two variables has infinitely many solutions.

Geometrically, each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa.

A linear equation in two variables alone can not be solved to find the particular value of x & y.

Let’s observe a statement,

Cost of two pencils & a pen is Rs 20.

We can express the given statement as an equation as

2x + y = 20

(Where, cost of a pencil & a pen be x & y respectively)

Can we solve this equation to find the cost of the pencil and pen?

We can find many possible solutions like (0,20), (1,18), (2,16)… but we can’t determine that particular pair of x & y which represent the cost of the pen & pencil respectively.

The data is insufficient here, we need some more information to solve the problem.

Let’s observe these statements

Cost of two pencils and a pen is Rs 20. Cost of the pen is twice the cost of the pencil.

Let, the cost of a pencil & a pen be x & y respectively

From the 1st statement

2+ y = 20 …(i)

From the 2nd statement

y = 2x

y − 2x = 0 …(ii)

Now, we have a pair of linear equations in two variables

let’s substitute y = 2x in eq (i), then

⇒ 2x + 2x = 20

⇒ 4x = 20

x = 20/4

x = 5

then, corresponding value of y

= 2x = 2×5 = 10

Thus, to solve a linear equation in two variables we need pair of them.

The general form for a pair of linear equations in two variables x and y is

a_1 x + b_1 y + c_1 = 0 \\  a_2 x + b_2 y + c_2 = 0,\,

where a_1, b_1, c_1, a_2, b_2, c_2 are all real numbers such that

a_1 ^2 + b_1 ^2 \neq 0, a_2 ^2 + b_2 ^2 \neq 0

Geometrical representation of pair of linear equations

Geometrical (i.e., graphical) representation of a linear equation in two variables is a straight line. Similarly, graph of a pair of linear equations in two variables will be two straight lines, both to be considered together.

Given two lines in a plane, only one of the following three possibilities can happen:

(i) The two lines will intersect at one point.

(ii) The two lines will not intersect, i.e., they are parallel.

(iii) The two lines will be coincident.

(i) (ii) (iii)

Solution of a Pair of Linear Equation

Among many solutions of individual equations of a pair, the solution in common to both equations is called the solution of the pair of linear equations.

Consider a pair of linear equations –

x – 2y = 0 and

3x + 4y = 20

Solution of x−2y = 0 are (0,0), (2,1), (4,2)…

Solution of 3+ 4y = 20 are (0,5), (4,2), (8,-1)…

(4,2) is common solution of both equations

∴ Solution of the given pair of linear equations is (4,2)

Hence, a solution of a pair of linear equations satisfies both equations of the pair.

It can be found both geometrically or algebraically.

Graphical Method of Solution of a Pair of Linear Equations

A linear equation is graphically represented by a line. To draw the line we need at least two points corresponding to two solution of the equation.

To represent a pair of linear equation in two variables we find two solutions of each equation.

We plot these points in a graph paper and draw the two lines representing each equation.

Case I: Two lines intersecting at a point

Let’s represent the given pair of equations graphically

x – 2y = 0 …(i)

3x + 4y = 20 …(ii)

Solution table of x − 2y = 0

x
0
2
y
0
1

Plot the points A(0,0) & B(2,1) on the graph & join them to obtain the line representing x − 2y = 0.

Solution table of 3x + 4y = 20

x
0
4
y
5
2

Plot the points C(0,5) & D(4,2) on the graph & join them to obtain the line representing 3x + 4y = 20.

The two lines are intersecting at the point (4, 2).

Therefore, the point (4, 2) lies on both the lines representing the equations x – 2y = 0 and 3x + 4y = 20. This implies that (4,2) is a common solution and hence satisfies both equations.

Thus, x = 4, y = 2 is a solution of the given pair of linear equations.

Let us verify algebraically that x = 4, y = 2 is a solution of the given pair of equations.

Substituting the values of x and y in each equation, we get

4 – 2 × 2 = 0

LHS = RHS

3(4) + 4(2) = 20

LHS = RHS

This verifies that x = 4, y = 2 is a solution of both the equations.

Since (4, 2) is the only common point on both the lines, there is one and only one solution for this pair of linear equations in two variables.

Case II: Two lines coincident to each other

Let’s consider following pair of equations

2x + 3y = 9 …(i)

4x + 6y = 18 …(ii)

Solution table of 2x + 3y = 9

x
-3
0
y
5
3

Plot the points A(-3,5) & B(0,3) on the graph & join them to obtain the line representing 2x + 3y = 9.

Solution table of 4x + 6y = 18

x
0
3
y
3
1

Plot the points B(0,3) & C(3,1) on the graph & join them to obtain the line representing 4x + 6y = 18.

From the graph, we observe that both the lines coincide and hence every point on the line is a common solution to both the equations.

Therefore, the equations 2x + 3y = 9 and 4x + 6y = 18 have infinitely many solutions.

This is so, because, both the equations are equivalent, i.e., one can be derived from the other. If we divide the equation 4x+ 6y = 18 by 2 , we get 2x + 3y = 9, which is the same as equation (i).

Case III: Two lines parallel to each other

Let’s consider following pair of equations

x + 2y – 4 = 0 …(i)

2x + 4y – 12 = 0 …(ii)

Solution table of x + 2y – 4 = 0

x
0
2
y
2
1

Plot the points A(0,2) & B(2,1) on the graph & join them to obtain the line representing x + 2y – 4 = 0.

Solution table of 2x + 4y – 12 = 0

x
0
2
y
3
2

Plot the points C(0,3) & D(2,2) on the graph & join them to obtain the line representing 2x + 4y – 12 = 0.

Here, we observe that the lines do not intersect anywhere, i.e., they are parallel.

This implies that the equations have no common solution.

Therefore, the equations x + 2y – 4 = 0 and 2x + 4y – 12 = 0 have no solutions.

Nature of a Pair of Linear Equations

Inconsistent pair of linear equations

A pair of linear equations which has no solution, is called an inconsistent pair of linear equations.

The graph of such pair has two parallel lines.

Consistent pair of linear equations.

A pair of linear equations in two variables, which has a solution, is called a consistent pair of linear equations.

Independent pair of linear equations

Independent pair of linear equations has a unique solution.

The graph of such pair has two intersecting lines.

Dependent pair of linear equations

A dependent pair of linear equations are equivalent and has infinitely many distinct common solutions.

The graph of such pair has two coincident lines.

Comparing Coefficients of a Pair of Linear Equations in Standard Form

We can Predict the nature of a pair of linear equations by comparing the coefficients of their terms in standard form.

a_1 x + b_1 y + c_1 = 0 \\ \, a_2 x + b_2 y + c_2 = 0,

(i)\, If\, \frac{a_1}{a_2} \neq \frac{b_1}{b_2}

then, the pair of linear equation is consistent.

(ii)\, If\, \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

then, the pair of linear equation is consistent and dependent.

(iii)\, If\, \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}

then, the pair of linear equation is inconsistent.

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