CBSE Class 9 Maths Chapter 4 Linear Equations in Two Variables NCERT Solutions

A linear equation in two variables, such as $2x\; -\; 5y\; =\; 7$, has the general form $Ax\; +\; By\; +\; C\; =\; 0$. For any such equation where A and B are not both zero, there are infinitely many pairs of values for x and y that satisfy the equation. This is because we can choose any real number for one variable and then solve for the other, resulting in an infinite number of possible solutions. Therefore, the correct option is (C) Infinitely many solutions.

A linear equation in two variables typically has infinitely many solutions when x and y can be any real numbers. However, if the domain for x and y is restricted, the number of solutions can be limited. If x and y are restricted to be natural numbers (positive integers), the possible pairs (x, y) that satisfy the equation $2x\; +\; 5y\; =\; 7$ become finite. For example, if y=1, then $2x\; =\; 2$, so $x=1$. If y is any other natural number, x will not be a natural number. Thus, restricting x and y to natural numbers can lead to a unique solution. Hence, the correct option is (A) Natural numbers.

We are given that the point (2, 0) is a solution to the linear equation $2x\; +\; 3y\; =\; k$. This means that when $x\; =\; 2$ and $y\; =\; 0$, the equation must hold true. Substitute these values into the equation:

$2\; \backslash times\; (2)\; +\; 3\; \backslash times\; (0)\; =\; k$

Performing the multiplication:

$4\; +\; 0\; =\; k$

Therefore, the value of k is 4.

Hence, the correct option is (A).

Consider the given linear equation in two variables:

$$2x\; +\; 0y\; +\; 9\; =\; 0$$

Since the coefficient of y is 0, the term $0y$ is always 0, regardless of the value of y. The equation simplifies to:

$2x\; +\; 9\; =\; 0$

Now, we solve for x:

$2x\; =\; -9$

$x\; =\; -\backslash frac\{9\}\{2\}$

This means that for any solution (x, y) of this equation, the value of x must be $-9/2$. The value of y can be any real number. We can represent any real number with the variable 'm'. Therefore, any solution is of the form $\backslash left(-\backslash frac\{9\}\{2\},\; m\backslash right)$.

Hence, the correct option is (A).

The graph of a linear equation cuts the y-axis at the point where the x-coordinate is zero. To find this point for the equation $2x\; +\; 3y\; =\; 6$, we substitute $x\; =\; 0$ into the equation:

$$2x\; +\; 3y\; =\; 6$$

Substitute $x\; =\; 0$:

$$2\; \backslash times\; (0)\; +\; 3\; \backslash times\; y\; =\; 6$$

This simplifies to:

$0\; +\; 3y\; =\; 6$

$3y\; =\; 6$

Now, solve for y:

$y\; =\; \backslash frac\{6\}\{3\}$

$y\; =\; 2$

So, the point where the graph cuts the y-axis is (0, 2).

Hence, the correct option is (D).

The given equation is $x\; =\; 7$. To express this equation in terms of two variables (x and y), we need to include both variables in the standard form $Ax\; +\; By\; +\; C\; =\; 0$ or $Ax\; +\; By\; =\; C$. In the equation $x\; =\; 7$, the coefficient of x is 1, and there is no y term explicitly written. This means the coefficient of y is 0. Therefore, we can write the equation as:

$1\; \backslash cdot\; x\; +\; 0\; \backslash cdot\; y\; =\; 7$

This form explicitly shows that the equation holds true for any value of y, as long as x is equal to 7. This represents a vertical line passing through x=7 on the x-axis.

Hence, the correct option is (B).