AcademicMathematicsNCERTClass 10 To do: We have to find out whether the given pairs of linear equations are consistent or inconsistent. Solution: (i) Given equations are: $3x + 2y=5;\ 2x – 3y=7$ $\frac{a_1}{a_2}=\frac{3}{2}$ $\frac{b_1}{b_2}=\frac{-2}{3}$ $\frac{c_1}{c_2}=\frac{5}{7}$ Here we find, $\frac{a_1}{a_2}≠\frac{b_1}{b_2}$ Thus, these linear equations are intersecting each other at only one point and they have only one possible solution. Therefore, the pair of linear equations is consistent. (ii) Given equations are: $2x-3y=8;\ 4x-6y=9$ $\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2}$ $\frac{b_1}{b_2}=\frac{-3}{-6}=\frac{1}{2}$ $\frac{c_1}{c_2}=\frac{8}{9}$ Here we find, $\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}$ Therefore, these linear equations are parallel to each other and thus have no possible solution. Thus, the pair of linear equations is inconsistent. (iii) Given equations are: $\frac{3x}{2}+\frac{5y}{3}=7;\ 9x-10y=14$. $\frac{a_1}{a_2}=\frac{\frac{3}{2}}{9}=\frac{1}{6}$ $\frac{b_1}{b_2}=\frac{\frac{5}{3}}{-10}=-\frac{1}{6}$ $\frac{c_1}{c_2}=\frac{7}{14}=\frac{1}{2}$ Here we find, $\frac{a_1}{a_2}≠\frac{b_1}{b_2}$ Thus, these linear equations are intersecting each other at only one point and they have only one possible solution. Therefore, the pair of linear equations is consistent. (iv) Given equations are: $5x-3y=11;\ -10x+6y=-22$ $\frac{a_1}{a_2}=\frac{5}{-10}=-\frac{1}{2}$ $\frac{b_1}{b_2}=\frac{-3}{6}=-\frac{1}{2}$ $\frac{c_1}{c_2}=\frac{11}{-22}=-\frac{1}{2}$ Here we find, $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ Therefore, these linear equations are coincident pairs of lines and thus have an infinite number of possible solutions. Thus, the given pair of linear equations is consistent. (v) Given equations are: $\frac{4x}{3}+2y=8;\ 2x+3y=12$ $\frac{a_1}{a_2}=\frac{\frac{4}{3}}{2}=\frac{2}{3}$ $\frac{b_1}{b_2}=\frac{2}{3}$ $\frac{c_1}{c_2}=\frac{8}{12}=\frac{2}{3}$ Here we find, $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ Therefore, these linear equations are coincident pairs of lines and thus have an infinite number of possible solutions. Therefore, the pair of linear equations is consistent.
Updated on 10-Oct-2022 13:19:43 |