Check whether the following equations are consistent or inconsistent solve them graphically 3x+2y=5

Solution:

For any pair of linear equation

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Consistent means pair of linear equations will have an unique solution or infinitely many solutions.

a₁/a₂ ≠ b₁/b₂ (Intersecting lines / unique Solution)

a₁/a₂ = b₁/b₂ = c₁/c₂ (Coincident Lines / Infinitely many Solutions)

a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (Parallel lines / No Solution)

(i) 3x + 2y = 5; 2x - 3y = 7

a₁/a₂ = 3/2

b₁/b₂ = 2/(-3)

c₁/c₂ = - 5/(-7) = 5/7

From the above,

a₁/a₂ ≠ b₁/b₂

Therefore, lines are intersecting and have a unique solution,

Hence, the pair of equations is consistent.

(ii) 2x - 3y = 8; 4x - 6y = 9

a₁/a₂ = 2/4 = 1/2

b₁/b₂ = -3/(-6) = 1/2

c₁/c₂= -8/(-9) = 8/9

From the above,

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Therefore, these lines are parallel and have no solution,

Hence, the pair of equations is inconsistent.

(iii) 3/2x + 5/3y = 7; 9x -10y = 14

a₁/a₂ = (3/2)/9 = (3/2) × (1/9) = 1/6

b₁/b₂ = (5/3)/(-10) = (5/3) × 1/(-10) = 1/(-6) = -1/6

From the above,

a₁/a₂ ≠ b₁/b₂

Therefore, lines are intersecting and have a unique solution.

Hence, they are consistent.

(iv) 5x - 3y = 11; -10x + 6y = -22

a₁/a₂ = 5/(-10) = -1/2

b₁/b₂ = -3/6 = -1/2

c₁/c₂ = -11/22 = -1/2

From the above,

a₁/a₂ = b₁/b₂ = c₁/c₂

Therefore, lines are coincident and have infinitely many solutions.

Hence, they are consistent.

(v) 4/3x + 2y = 8; 2x + 3y = 12

a₁/a₂ = (4/3)/2 = (4/3) × (1/2) = 2/3

b₁/b₂ = 2/3

c₁/c₂= -8/(-12) = 2/3

From the above,

a₁/a₂ = b₁/b₂ = c₁/c₂

Therefore, lines are coincident and have infinitely many solutions.

Hence, they are consistent.

☛ Check: Class 10 Maths NCERT Solutions Chapter 3

Video Solution:

On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, find out whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2 y = 5; 2x - 3y = 7 (ii) 2x - 3y = 8; 4x - 6 y = 9 (iii) 3/2x + 5/3y = 7; 9x -10y = 14 (iv) 5x - 3y = 11; -10x + 6 y = -22 (v) 4/3x + 2 y = 8; 2x + 3y = 12

NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.2 Question 3

Summary:

On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, we have seen whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2 y = 5; 2x - 3y = 7 consistent. (ii) 2x - 3y = 8; 4x - 6 y = 9 inconsistent. (iii) 3/2x + 5/3y = 7; 9x -10y = 14 consistent. (iv) 5x - 3y = 11; -10x + 6 y = -22 consistent. (v) 4/3x + 2 y = 8; 2x + 3y = 12 consistent.

☛ Related Questions:

Mathematics Class 10>Pair of Linear Equations in Two Variables>Exercise>Q 2

1. General Form of A Pair Of Linear Equations In Two Variables:

A pair of linear equations in two variables x and y can be represented algebraically as follows:

a1x+b1y+c1=0

a2x+b2y+c2=0, where a1, a2, b1, b2, c1, c2 are real numbers such that a12+b12≠0, a22+b22≠0.

2. Graphical or Geometrical Meaning of a Pair of Linear Equations:

a1x+b1y+c1=0 and a2x+b2y+c2=0 are two equations representing a pair of straight lines which are

(i) intersecting, if a1a2≠b1b2

(ii) parallel, if a1a2=b1b2≠c1c2

(iii) coincident, if a1a2=b1b2=c1c2

3. Graphical Method of Solving Linear Pair of Equations:

To solve a pair of linear equations in two variables by the Graphical method, we first draw the lines represented by them.

(i) If the pair of lines intersect at a point, then we say that the pair is consistent, and the coordinates of the intersection point provide us the unique solution.

(ii) If the pair of lines are parallel, then the pair has no solution and is called an inconsistent pair of equations.

(iii) If the pair of lines are coincident, then it has infinitely many solutions. Each point on the line is a solution. In this case, we say that the pair of linear equations is consistent with infinitely many solutions.

4. Algebraic Method of Solving Linear Pair of Equations:

To solve a pair of linear equations in two variables algebraically, we have the following methods:

(i) Substitution method

(ii) Elimination method

(iii) Cross-multiplication method

5. Nature of Solution of Linear Pair of Equations:

If a1x+b1y+c1=0 and a2x+b2y+c2=0 is a pair of linear equations in two variables x and y such that

(i) a1a2≠b1b2 the pair of linear equations is consistent with a unique solution

(ii) a1a2=b1b2≠c1c2 then the pair of linear equations is inconsistent and has no solution.

(iii) a1a2=b1b2=c1c2 then the pair of linear equations is consistent with infinitely many solutions.

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