Answer  Hint: The system of linear equations in two variables are said to be consistent if the set of equations have at least one solution and called as inconsistent if they have no solution at all. The consistency of the solution is determined by comparing the ratios of coefficient of variables and the constant term. Graphs of any equation could be drawn by using at least two coordinates. Complete step by step answer: We are given the two equations and we have to find whether it is consistent or inconsistent. Also we have to draw the graph of the two equations. First of all a set of two equation \[{a_1}x + {b_1}y + {c_1} = 0\] and \[{a_2}x + {b_2}y + {c_2} = 0\] are said to be consistent if the equation has at least one solution and is said to be inconsistent if the equations have no solution at all.We have three conditions for analysing the consistency of the equations which is as follows:-If \[\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}\] The pair of equation is consistent and have unique solution (only one solution).-If \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\] The pair for equation is consistent and have infinitely many solutions.-If \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}\] The pair of equation is inconsistent and have no solutions at all.Here from the equation \[2x - 2y - 2 = 0\] and \[4x - 4y - 5 = 0\]We have \[{a_1} = 2, {b_1} = - 2,{c_1} = - 2\]\[\Rightarrow {a_2} = 4, {b_2} = - 4, {c_2} = - 5\]Here,\[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{4} = \dfrac{1}{2},\, \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{ - 2}}{{ - 4}}{\text{ = }}\dfrac{1}{2}{\text{,}} \dfrac{{{c_1}}}{{{c_2}}} = \dfrac{{ - 2}}{{ - 5}}\]Clearly we see that \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}\]Therefore the pair of linear equations is INCONSISTENT.Now we have to draw the graph of both the equationFor \[2x - 2y - 2 = 0\]We have the given set of points satisfying the equation We have obtained the values by putting one of the coordinates as zero.
Note:Graphs need at least two points to be drawn. Consistent solutions with infinite solutions have the same slope for both the equations. Consistent equations with only one solution have the different slopes which intersect only once. Inconsistent solutions have the slope of the graph parallel to each other for both equations.
2
x + y = 5 2x + 2y = 10
Here, a1 = 1, b2 = 1, c1 = 5 a1 = 2, b2 = 2, c1 = 10 ∴
∴ two lines are inconsistent
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: 1. Linear Equation in Two Variables: (i) A pair of linear equation in two variables is expressed as ax+by+c=0, where a, b, c∈R and a≠0 and b≠0. 2. Simultaneous Linear Equations: (i) When we consider two linear equations in two variables having unique set of solution, such equations are called simultaneous equations. (ii) Two linear equations in the same two variables are called a pair oflinear equations in two variables. (iii) When the simultaneous equations are satisfied by the same pair of values of the two variables, then such pair of values of the variables is called the solution of the given simultaneous equations. 3. A pair of linear equations in two variables can be solved by the: (i) Graphical method (ii) Algebraic method. 4. To solve a pair of linear equations in two variables by 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 point provide us the unique solution. (ii) If the pair of lines are parallel, then the pair has no solution and is called inconsistent pair of equations. (iii) If the pair of lines are coincident, then it has infinitely many solutions each point on the line being of solution. In this case, we say that the pair of linear equations is consistent with infinitely many solutions. 3. Methods of solving pair of linear equations in two variables algebraically: (i) Substitution method. (ii) Cross-multiplication method. (iii) Elimination method. 4. Consistency in Linear Equations in Two Variables: (i) The linear equations in two variables are said to be consistent if they have solution. (ii) The linear equations in two variables are said to be inconsistent if they do not have solution. 5. Condition of Consistency: For the given equations a1x+b1y+c1=0 and a2x+b2y+c2=0 the nature of solution depends upon the following three conditions : (i) If a1a2≠b1b2, then the solution is unique. Equations are consistent. The lines are intersecting lines. (ii) If a1a2=b1b2≠c1c2, then there is no solution. Equations are not consistent. The lines are parallel lines. (iii) If a1a2=b1b2=c1c2, then there are infinite solutions. Equations are consistent. The lines are coincident lines. 6. There are several situations which can be mathematically represented by two equations that are not linear to start with. But we can alter them so that they will be reduced to a pair of linear equations.
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Check whether the following equations are consistent or inconsistent solve them graphically x+y=5
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