Given pair of equations \[\frac{3}{2}\]x + \[\frac{5}{3}\]y = 7 and 9x – 10y = 12
Now take \[\frac{3}{2}\]x + \[\frac{5}{3}\]y = 7
⇒ \[\frac{9x+10y}{6}\]= 7
⇒ 9x + 10y = 42 and 9x – 10y =12
a1/a2 = 9/9 = 1/1
b1/b2 = 10/-10 = 1/-1 and
c1/c2 = -42/-12 = 7/2
Since \[\frac{a_1}{a_2}\] ≠ \[\frac{b_1}{b_2}\] they are intersecting lines and hence consistent pair of linear equations.
Solution: The unique solution of given pair of equations is [3.1, 1.4]
Solution:
For any pair of linear equation,
a₁ x + b₁ y + c₁ = 0
a₂ x + b₂ y + c₂ = 0
a] a₁/a₂ ≠ b₁/b₂ [Intersecting Lines/uniqueSolution]
b] a₁/a₂ = b₁/b₂ = c₁/c₂ [Coincident Lines/Infinitely many Solutions]
c] a₁/a₂ = b₁/b₂ ≠ c₁/c₂ [Parallel Lines/No solution]
[i] x + y = 5, 2x + 2y = 10
a₁/a₂= 1/2
b₁/b₂= 1/2
c₁/c₂= -5/[-10] = 1/2
From the above,
a₁/a₂ = b₁/b₂ = c₁/c₂
Therefore, lines are coincident and have infinitely many solutions. Hence, they are consistent.
x + y - 5 = 0
y = - x + 5
y = 5 - x
2x + 2y - 10 = 0
2y = 10 - 2x
y = 5 - x
All the points on coincident line are solutions for the given pair of equations.
[ii] x - y = 8, 3x - 3y =16
a₁/a₂ = 1/3
b₁/b₂ = -1/[-3] = 1/3
c₁/c₂ = - 8/[-16] = 1/2
From the above,
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Therefore, lines are parallel and have no solution.
Hence, the pair of equations are inconsistent.
[iii] 2x + y - 6 = 0, 4x - 2y - 4 = 0
a₁/a₂ = 2/4 = 1/2
b₁/b₂ = 1/[-2] = -1/2
c₁/c₂ = -6/[-4] = 3/2
From the above,
a₁/a₂ ≠ b₁/b₂
Therefore, lines are intersecting and have a unique solution.
Hence, they are consistent.
2x + y - 6 = 0
y = 6 - 2x
4x - 2y - 4 = 0
2y = 4x - 4
y = 2x - 2
x = 2 and y = 2 are solutions for the given pair of equations.
[iv] 2x - 2y - 2 = 0, 4x - 4y - 5 = 0
a₁/a₂ = 2/4 = 1/2
b₁/b₂ = -2/[-4] = 1/2
c₁/c₂ = -2/[-5] = 2/5
From the above,
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Therefore, lines are parallel and have no solution.
Hence, the pair of equations are inconsistent.
☛ Check: NCERT Solutions for Class 10 Maths Chapter 3
Video Solution:
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: [i] x + y = 5, 2x + 2y = 10 [ii] x - y = 8, 3x - 3y =16 [iii] 2x + y - 6 = 0, 4x - 2y - 4 = 0 [iv] 2x - 2y - 2 = 0, 4x - 4y - 5 = 0
NCERT Solutions for Class 10 Maths - Chapter 3 Exercise 3.2 Question 4
Summary:
On comparing the ratios of the coefficients of the following pairs of linear equations, we see that [i] x + y = 5, 2x + 2y = 10 have infinitely many solutions. Hence, they are consistent. [ii] x - y = 8, 3x - 3y =16 are parallel and have no solution.Hence, the pair of equations are inconsistent. [iii] 2x + y - 6 = 0, 4x - 2y - 4 = 0 are intersecting and have a unique solution. Hence, they are consistent. [iv] 2x - 2y - 2 = 0, 4x - 4y - 5 = 0 are parallel and have no solution. Hence, the pair of equations are inconsistent.
☛ Related Questions:
- Form the pair of linear equations in the following problems and find their solutions graphically. [i] 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz. [ii] 5 pencils and 7 pens together cost ₹ 50, whereas 7 pencils and 5 pens together cost ₹ 46. Find the cost of one pencil and that of one pen.
- On comparing the ratios a1/a2 = b1/b2 = c1/c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: i] 5x – 4y + 8 = 0 7x + 6y – 9 = 0 [ii] 9x + 3y + 12 = 0 18x + 6y + 24 = 0 [iii] 6x – 3y + 10 = 0 2x – y + 9 = 0
- On comparing the ratios find out whether the following pair of linear 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