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# X Class - CBSE Nature of system of linear equations : Let $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y+c_{2}=0$ is a system of | S.No. | Compare <br> ratios | Alachvale <br> Interpretation | Graphical <br> Representation | Nature of the <br> System | | :-- | :-- | :-- | :-- | :-- | | 1. | $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ | Exactly one <br> solution (unique) | Intersecting lines | Consistent | | 2. | $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{2}}{c_{2}}$ | Infinitely many <br> solutions | Coincident lines | Consistent and dependent | | 3. | $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{2}}{c_{2}}$ | No solution | Parallel lines | Inconsistent | Note: The solution of a consistent system of linear equations is the coordinates of intersecting point of the two lines represented by them. ## EXERCISE 3.1 1. On comparing the ratios, $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}$ and $\frac{c_{1}}{c_{2}}$, find out whether the lines representing the given pairs of linear equations intersect a point, are parallel or coincident. Sol. i) $5 x-4 y+8=0$ $7 x+6 y-9=0$ Comparing the given equations with standard forms of equations $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y+c_{2}=0$ we have $$ a_{1}=5, b_{1}=-4, c_{1}=8 ; a_{2}=7, b_{2}=6, c_{2}=-9 $$ Now $\frac{a_{1}}{a_{2}}=\frac{5}{7}, \frac{b_{1}}{b_{2}}=\frac{-4}{6}$ $$ \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} $$ Thus the lines representing the pair of linear equation are intersecting. ii) $9 x+3 y+12=0,18 x+6 y+24=0$ We have $a_{1}=9, b_{1}=3, c_{1}=12 ; a_{2}=18, b_{2}=6, c_{2}=24$ Now $\frac{a_{1}}{a_{2}}=\frac{9}{18}=\frac{1}{2}, \frac{b_{1}}{b_{2}}=\frac{3}{6}=\frac{1}{2}$ and $\frac{c_{1}}{c_{2}}=\frac{12}{24}=\frac{1}{2}$

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