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 = 0NCERT 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: Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically: x – y = 8, 3x – 3y = 16 x - y = 8 3x - 3y = 16 `a_1/a_2 = 1/3, b_1/b_2 = (-1)/-3 = 1/3 and c_1/c_2 = 8/16 = 1/2` Since `a_1/a_2 = b_1/b_2 ≠ c_1/c_2` Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent. Concept: Graphical Method of Solution of a Pair of Linear Equations Is there an error in this question or solution?
The coach of a cricket team buys 3 bats and 6 balls for Rs. 3900. Later, she buys another bat and 3 more balls of the same kind for Rs. 1300. Represent this situation algebraically and graphically.
Let the cost of 1 bat be Rs. x and cost of I ball be Rs.y Case I. Cost of 3 bats = 3x Cost of 6 balls = 6y According to question, 3x + 6y = 3900 Case II. Cost of I bat = x Cost of 3 more balls = 3y According to question, x + 3y = 1300 So, algebraically representation be 3x + 6y = 3900 x + 3y = 1300 Graphical representation : We have, 3x + 6y = 3900 ⇒ 3(x + 2y) = 3900 ⇒ x + 2y = 1300 ⇒ a = 1300 - 2y Thus, we have following table : We have, x + 3y = 1300 ⇒ x = 1300 - 3y Thus, we have following table : When we plot the graph of equations, we find that both the lines intersect at the point (1300. 0). Therefore, a = 1300, y = 0 is the solution of the given system of equations. |
Which of the following Pair of linear equations are consistent or inconsistent xy 8, 3x -- 3y = 16
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