Linear Equations And Inequalities In Two Variables

For any pair of linear equations to be consistent or inconsistent, we

check the following situations:

a.

In this case, the pair of linear equations is consistent. This means there is unique solution for the given pair of linear equations. The graph of linear equations would be two intersecting lines.

b.

In this case, the pair of linear equations is inconsistent. This means there is no solution for the given pair of linear equations. The graph of linear equations will be two parallel lines.

c.

In this case, the pair of linear equations is dependent and consistent. This means there are infinitely many solutions for the given pair of linear equations. The graph of linear equations will be coincident lines.

(i) 2x – 3y = 8; 4x – 6y = 9

Here, a₁ = 2, b₁ = – 3, c₁ = 8

and a₂ = 4, b₂ = – 6, c₂ = 9

Clearly,

Hence, the given lines are parallel. So the given pair of equation has no solution and it is inconsistent.

(ii) 3x – y = 2; 6x – 2y = 4

Here, a₁ = 3, b₁ = – 1, c₁ = 2

and a₂ = 6, b₂ = – 2, c₂ = 4

Clearly,

Hence, the given lines are dependent and consistent. So the given pair of equation has infinitely many solutions.

(iii) 2x – 2y = 2; 4x – 4y = 5

Here, a₁ = 2, b₁ = – 2, c₁ = 2

and a₂ = 4, b₂ = – 4, c₂ = 5

Clearly,

Hence, the given lines are parallel. So the given pair of equation has no solution and it is inconsistent.

(iv)

Here, , b₁ = 2, c₁ = 8

and a₂ = 2, b₂ = 3, c₂ = 12

Clearly,

Hence, the given lines are two intersecting lines. So the given pair of equation has unique solution and it is consistent.

For a given pair of linear equations in two variables, the graph is represented by two lines.

● If the lines intersect at a point, that point gives the unique solution for the two equations. If there is a unique solution of the given pair of equations, the equations are called consistent.

● If the lines coincide, there are indefinitely many solutions for the pair of linear equations. In this case, each point on the line is a solution. If there are infinitely many solutions of the given pair of linear equations, the equations are called dependent (consistent).

● If the lines are parallel, there is no solution for the pair of linear equations. If there is no solution of the given pair of linear equations, the equations are called inconsistent.

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For x + y = 3

y = 3– x

For 3x – 2y = 4

As seen from the graph, the two lines each other at intersecting point

(2,1). Hence, roots are consistent with unique solution.

For a given pair of linear equations in two variables, the graph is represented by two lines.

● If the lines intersect at a point, that point gives the unique solution for the two equations. If there is a unique solution of the given pair of equations, the equations are called consistent.

● If the lines coincide, there are indefinitely many solutions for the pair of linear equations. In this case, each point on the line is a solution. If there are infinitely many solutions of the given pair of linear equations, the equations are called dependent (consistent).

● If the lines are parallel, there is no solution for the pair of linear equations. If there is no solution of the given pair of linear equations, the equations are called inconsistent.

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For 2x – y = 4

y = 2x – 4

For x + y = –1

Y = – 1 – x

As seen from the graph, the two lines each other at intersecting point

(1, – 2). Hence, roots are consistent with unique solution.

For a given pair of linear equations in two variables, the graph is represented by two lines.

● If the lines intersect at a point, that point gives the unique solution for the two equations. If there is a unique solution of the given pair of equations, the equations are called consistent.

● If the lines coincide, there are indefinitely many solutions for the pair of linear equations. In this case, each point on the line is a solution. If there are infinitely many solutions of the given pair of linear equations, the equations are called dependent (consistent).

● If the lines are parallel, there is no solution for the pair of linear equations. If there is no solution of the given pair of linear equations, the equations are called inconsistent.

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For x + y = 5

y = 5 – x

For 2x + 2y = 10

2y = 10 – 2x

As seen from the graph and the table, the two lines are parallel and

overlap each other. Hence, roots of the equations are consistent or

dependent with infinitely many solutions.

For a given pair of linear equations in two variables, the graph is represented by two lines.

● If the lines intersect at a point, that point gives the unique solution for the two equations. If there is a unique solution of the given pair of equations, the equations are called consistent.

● If the lines coincide, there are indefinitely many solutions for the pair of linear equations. In this case, each point on the line is a solution. If there are infinitely many solutions of the given pair of linear equations, the equations are called dependent (consistent).

● If the lines are parallel, there is no solution for the pair of linear equations. If there is no solution of the given pair of linear equations, the equations are called inconsistent.

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For 3x + y = 2

y = 2 – 3x

For 2x – 3y = 5

3y = 2x – 5

As seen from the graph, the two lines each other at intersecting point

(1, – 1). Hence, roots of the two lines are consistent with unique solution.

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For 2x – 5y + 4 = 0

5y = 2x + 4

For 2x + y – 8 = 0

y = 8 – 2x

The coordinates of the points where the lines intersect the y – axis are: (0,0.8) and (0,8).

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For 3x + 2y = 12

2y = 12 – 3x

For 5x – 2y = 4

2y = 5x – 4

The coordinates of the points where the lines intersect the y – axis are: (0,6) and (0, – 2).

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For 4x – 5y = 20

5y = 4x – 20

For 3x + 5y = 15

5y = 3x – 15

The coordinates of the vertices of the triangle formed by lines and the

y – axis are: (0, – 4), (0,3) and (5,0).

To find the solution set for x ≥ 2 graphically, we first draw x = 2 which is a line parallel to y – axis (as seen from the graph). x>2 is represented as a shaded region on the right of x = 2, including all the points lying on the line x = 2.

To find the solution set for y ≤ –3 graphically, we first draw y = – 3 which

is a line parallel to x – axis (as seen from the graph). y< – 3 is represented

as a shaded region below y = – 3, including all the points lying on the

line y = – 3.

To find the solution set for x – 2y < 0 graphically, we first draw x – 2y = 0

which is a dotted line passing through the origin (as seen from the

graph). x – 2y < 0 is represented as a shaded region above x – 2y = 0,

excluding all the points lying on the line x – 2y = 0.

To find the solution set for 2x + 3y ≤ 6 graphically, we first draw 2x +

3y = 6,a line as shown in the graph.

As,

(0, 0) satisfies the inequality

2x + 3y < 6, hence

2x + 3y ≤ 6 is represented as a

shaded region below 2x + 3y = 6, including all the points lying on the

line 2x + 3y = 6.

To find the solution set for |x|≤ 3 graphically, we first draw x = 3

and x = – 3, two lines drawn parallel to x – axis.

Now,

|0|< 3, satisfies the inequality hence

|x|≤ 3 is represented as

a shaded region between x = 3 and x = – 3, including all the points lying on

the lines x = 3 and x = – 3.

To find the solution set for 3x – 2y ≤ x + y – 8 graphically, we first

draw the line 3x – 2y = x + y – 8, as shown in the graph. 3x – 2y ≤ x +

y – 8 is represented as a shaded region above the line, including all the

points lying on the line 3x – 2y = x + y – 8.

To find the solution set for |x – y|≥1 graphically, we first draw x – y = 1

and x – y = – 1, two lines drawn parallel origin. |x – y|≥1 is represented

as a shaded region above and below x – y = 1 and x – y = – 1, including all the

points lying on the lines x – y = 1 and x – y = – 1.

A pair of linear equations has no solution when,

Given, x + y – 4 = 0

2x + ky – 3 = 0

Here, a₁ = 1, b₁ = 1, c₁ = 4

and a₂ = 2, b₂ = k, c₂ = 3

∴

∴k = 2

A pair of linear equations has no solution when,

Given, 3x – y = 0

kx + 5y = 0

Here, a₁ = 3, b₁ = – 1, c₁ = 0

and a₂ = k, b₂ = 5, c₂ = 0

∴

A pair of linear equations has no solution when,

Given, kx – y = 2

6x – 2y = 3

Here, a₁ = k, b₁ = – 1, c₁ = 2

and a₂ = 6, b₂ = – 2, c₂ = 3

∴

As evident from the graph, the inequations x ≥ 0, y ≥ 0 are overlapping

represent x and y axis both.

As can be seen from the graph inequation y – 3 ≤ 0 is parallel to x – axis.

Here, a₁ = 1, b₁ = 2, c₁ = 8

and a₂ = 2, b₂ = 4, c₂ = 16

This means there are infinitely many solutions for the given pair of

linear equations.

Given, 2x + 3y = 7; (a + b)x + (2a – b)y = 21

We know that a pair of linear equations has infinitely many solutions

when,

Here, a₁ = 2, b₁ = 3, c₁ = 7

and a₂ = (a + b), b₂ = (2a – b), c₂ = 21

A + b = 6 …(i)

2a – b = 9 …(ii)

Adding both the equations we get,

3a = 15

∴a = 5

∴b = 6 – 5

b = 1

To find the solution set for |x|≤ 3 graphically, we first draw x = 3

and x = – 3, two lines drawn parallel to x – axis. |x|≤ 3 is represented as a shaded region between x = 3 and x = – 3, including all the points lying on the lines x = 3 and x = – 3.

To find the solution set for 2x + 3y ≥ 3 graphically, we first

draw the line 2x + 3y = 3, as shown in the graph. 2x + 3y ≥ 3 is

represented as a shaded region above the line, including all the

points lying on the line 2x + 3y = 3.

Given, x = 3y = 6, 2x – 3y = 12 and 4x + 3y = a

∴ x = 6

Putting the value of x in 2x – 3y = 12, we get

2 × 6 – 3y = 12

12 – 3y = 12

– 3y = 0

∴ y = 0

Hence the value of a is: a = 4x + 3y

= 4 × 6 + 3 × 0

= 0

So the two linear equations are:

i. 2x – 3y = 12

ii. 4x + 3y = 24

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For 2x – 3y = 12

3y = 2x – 12

For 4x + 3y = 24

3y = 24 – 4x

Now plotting these two equations on graph,

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For 3x + 2y = 12

2y = 12 – 3x

For 5x – 2y = 4

2y = 5x – 4

The coordinates of the points where the lines represented by these intersect the y – axis are:

i. (0,6)

ii. (0, – 2)