Check whether the following equations are consistent or inconsistent 3/2x+5/3y=7
Given pair of equations \(\frac{3}{2}\)x + \(\frac{5}{3}\)y = 7 and 9x – 10y = 12 Show
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 = 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:
Is equations 3x 2y 5 and 2x 3y 7 are consistent or inconsistent?On comparing the ratio, (a1/a2) , (b1/b2) , (c1/c2) find out whether **3x + 2y = 5 ; 2x – 3y = 7** are consistent or inconsistent. So, the given equations intersect each other at one point and they have only one possible solution. The equations are consistent.
Which of the following pairs of linear equations are consistent inconsistent 3x 2y 5 2x 3y 7?3x + 2y = 5; 2x - 3y = 7. Hence, the given lines are intersecting. So, the given pair of linear equations has exactly one solution and therefore it is consistent.
Which of the following pairs of linear equations are consistent inconsistent 2x 3y 8 4x 6y 9?On comparing the ratio, (a1/a2) , (b1/b2) , (c1/c2) find out whether 2x – 3y = 8 ; 4x – 6y = 9 are consistent or inconsistent. So, the equations are parallel to each other and they have no possible solution. Hence, the equations are inconsistent.
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