Question

Class 9

1. Explain linear equation with suitable 2 examples . each example should be proper explaination .

2.Explain equations of lines parallel to the x-axis and y-axis with suitable 2 examples . each example should be proper explaination .

Explain correctly with good explanation with 2 suitable examples should be explained it correctly .

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Answer 1

A linear equation is simply a first degree equation of the form $a_1x_1+a_2x_2+a_3x_3+\,\dots\,+a_nx_n+b=0$ where $x_i$ are variables and $b,\ a_i$ are real constants, may or may not be zero, for all $i=1,\ 2,\ 3,\,\dots\,,\ n.$

The linear equations of the form $Ax+By+C=0$ can represent straight lines in the Cartesian plane having slope $-\dfrac{A}{B}$ and y intercept $-\dfrac{C}{B}.$

$\longrightarrow Ax+By+C=0$

$\longrightarrow \dfrac{A}{B}x+y+\dfrac{C}{B}=0$

$\longrightarrow y=-\dfrac{A}{B}x-\dfrac{C}{B}$

Comparing with slope - intercept equation of a straight line we get the slope is $-\dfrac{A}{B}$ and y intercept is $-\dfrac{C}{B}.$

Two examples are given below.

(1) $x+5y+3=0$

This equation represents a straight line having slope $-\dfrac{1}{5}$ and y intercept $-\dfrac{3}{5}.$ [Fig. (1)]

(2) $2x+4y-7=0$

This equation represents a straight line having slope $-\dfrac{2}{4}=-\dfrac{1}{2}$ and y intercept $\dfrac{7}{4}.$ [Fig. (2)]

The lines parallel to x axis have the equation of the form $y=k$ for some real constant $k,$ because such a line passes through all points having same y coordinate that is $k.$

Two examples are given.

(1) $y=3$

This equation represents a straight line parallel to x axis and passing through all points having y coordinate 3. [Fig. (3)]

(2) $y=-5$

This equation represents a straight line parallel to x axis and passing through all points having y coordinate -5. [Fig. (3)]

The lines parallel to y axis have the equation of the form $x=k$ for some real constant $k,$ because such a line passes through all points having same x coordinate that is $k.$

Two examples are given.

(1) $x=3$

This equation represents a straight line parallel to y axis and passing through all points having x coordinate 3. [Fig. (4)]

(2) $x=-5$

This equation represents a straight line parallel to y axis and passing through all points having x coordinate -5. [Fig. (4)]

Answer 2

1. A linear equation is an algebraic equation that denotes a straight line in a graph. It is a combination of constants and variables with the power of one. The no. of variables in a linear equation can vary and depending upon the the no. of variables in an equation they are named as:

✳ Linear equation in one variable

➡ A linear equation containing only one variable is termed as a linear equation in one variable. It is in the format ax + b = 0

✳ Linear equation in two variable etc.

➡ A linear equation containing two variables is termed as a linear equation in two variables. It is in the format ax + by + c =0.

Examples

$\sf{\longrightarrow\:5x+10=35}$

The linear equation consists only one variable "x" and hence it is a linear equation in one variable.

Here the value of x will be:

$\sf{=5x=35-10}$

$\sf{=5x=25}$

$\sf{=x=\dfrac{25}{5}}$

$\sf{=x=5}$

The value of x is 5.

$\sf{\longrightarrow\:5x+2y=8}$

The linear equation contains two variables "x" and "y" and hence it is a linear equation in two variables.

Here, by inspection, the value of x and y can be given as 0 and 4. Substituting those values:

$\sf{=(5\times0)+(2\times4)=8}$

The values of x and y are (0,4).

2. For a linear equation of two variables to be parallel to the x-axis, then y must be equal to k (y = k) and the k constant must be the same at all points whereas the x factor can have varying constants.

For example:

Giving the value of y = 3, we get a line passing through x = 0, x = 1 etc. and the line we get (with values (0,3), (1,3), (2,3), (3,3) etc.) will be parallel to the x axis.

[ Please refer the first attachment to view the diagram.]

For a linear equation of two variables to be parallel to the y - axis, then x must be equal to k (x = k) and the k constant must be the same at all points whereas the y factor can have varying constants.

For example:

Giving the value of x = 4, 23 get a line passing through y = 1, y = 2 etc. and the line we get (with values (4,1), (4,2), (4,3) etc.) will be parallel to the y axis.

[ Please refer the second attachment to view the diagram. ]