CBSE Class 10 Maths NCERT Solutions: Pair of Linear Equations in Two Variables

To determine the nature of the lines represented by the given pair of equations, we compare the ratios of their coefficients.

The given equations are:

6x - 3y + 10 = 0

2x - y + 9 = 0

Here, we have:

a_1 = 6, b_1 = -3, c_1 = 10

a_2 = 2, b_2 = -1, c_2 = 9

Now, let's find the ratios:

\frac{a_1}{a_2} = \frac{6}{2} = 3

\frac{b_1}{b_2} = \frac{-3}{-1} = 3

\frac{c_1}{c_2} = \frac{10}{9}

We observe that \frac{a_1}{a_2} = \frac{b_1}{b_2} but \frac{a_1}{a_2}

eq \frac{c_1}{c_2}.

According to the conditions for linear equations, when \frac{a_1}{a_2} = \frac{b_1}{b_2}

eq \frac{c_1}{c_2}, the pair of linear equations has no solution, and the lines represented are parallel.

Therefore, the correct option is (iv) parallel.

We need to determine the number of solutions for the given pair of linear equations by comparing their coefficients.

The equations are:

x + 2y + 5 = 0

-3x - 6y + 1 = 0

The coefficients are:

a_1 = 1, b_1 = 2, c_1 = 5

a_2 = -3, b_2 = -6, c_2 = 1

Let's calculate the ratios of the coefficients:

\frac{a_1}{a_2} = \frac{1}{-3}

\frac{b_1}{b_2} = \frac{2}{-6} = \frac{1}{-3}

\frac{c_1}{c_2} = \frac{5}{1}

Comparing these ratios, we find that \frac{a_1}{a_2} = \frac{b_1}{b_2} (since both are -1/3), but \frac{a_1}{a_2}

eq \frac{c_1}{c_2} (since -1/3

eq 5).

When \frac{a_1}{a_2} = \frac{b_1}{b_2}

eq \frac{c_1}{c_2}, the pair of linear equations has no solution. This means the lines represented by these equations are parallel and distinct.

Therefore, the pair of equations has no solution.

A pair of linear equations is called consistent if it has at least one solution. Let's consider the conditions for consistency:

1. Unique Solution: The lines intersect at exactly one point. This occurs when the ratio of the coefficients of x is not equal to the ratio of the coefficients of y. Mathematically, \frac{a_1}{a_2}

eq \frac{b_1}{b_2}. In this case, the lines are intersecting.

2. Infinitely Many Solutions: The lines are coincident, meaning they are the same line. This occurs when the ratios of all coefficients are equal. Mathematically, \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}. In this case, the lines are coincident.

An inconsistent system has no solution, which corresponds to parallel lines (\frac{a_1}{a_2} = \frac{b_1}{b_2}

eq \frac{c_1}{c_2}).

Since a consistent system implies either a unique solution (intersecting lines) or infinitely many solutions (coincident lines), the lines will be either intersecting or coincident.

Therefore, the correct option is (c) intersecting or coincident.

The first equation, $y\; =\; 0$, represents the x-axis in the Cartesian coordinate system. The second equation, $y\; =\; -7$, represents a horizontal line that is parallel to the x-axis and is located 7 units below it.

Parallel lines, by definition, never intersect. Since the solutions to a pair of linear equations correspond to the points where their graphs intersect, and these two lines do not intersect, there are no common points.

Therefore, the pair of equations $y\; =\; 0$ and $y\; =\; -7$ has no solution.

The correct option is (iv) no solution.