Linear Equations In Two Variables Class 9th Notes - Free NCERT Class 9th Maths Chapter 4 Notes - Download PDF

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Linear Equations in Two Variables Class 9 Notes

Linear Equations in One Variable: An equation that has one variable with a degree of one is called a linear equation in one variable.
The standard form of a linear equation in one variable is ax + b = 0, where a is not equal to zero.
Example: 2x + 3 = 0, 4x = 1

Linear Equations in Two Variables: An equation that has two variables with a degree of one is known as a linear equation in two variables.
The standard form of a linear equation in two variables is ax + by + c = 0, where a and b are not equal to zero.
Example: 3x + 2y = 5, 2a + 3b + 4 = 0

Establish a Linear Equation in Two Variables

The linear equation in two variables can be created by assuming the variables with the values as given in the example.
Example: The sum of the cost of a bat and a ball is 12. Write a linear equation in two variables to represent this statement.
Let the bat be represented by x and the ball by y.
According to the statements,
The equation is: x + y = 12

Solution of Linear Equations in Two Variables

The linear equation in two variables has two variables, which consist of two numbers and are known as the solution of the linear equation in two variables.
Some points related to the solution of linear equations in two variables:
1. Solution can be determined by the assumption method, and according to this method value of one variable is assumed, and this variable is replaced with this value and proceed to find the solution for the other variable is found.
2. There are many possible solutions for the same linear equation in two variables.

Graphical Representation of a Linear Equation in Two Variables

For any linear equation ax + by + c = 0, the pair of solutions in the form of variables (x, y) can be represented in the coordinate plane. With these coordinate values, the line doesn't always need to cut the coordinate axes.

Example: Write four solutions for the equation 3x + y = 6.

The graphical representation for the given coordinates is as follows,

Important Points:

1. Every coordinate in the graphical representation is the solution of a linear equation.
2. In a linear equation in two variables, when any number is added, multiplied, divided or subtracted from both sides of the equation, then the equation will be unaffected.

Lines Passing Through the Origin

When a linear equation has a solution coordinate (0, 0), and if these coordinates are represented in the coordinate plane, then these points pass through the origin, as shown in the figure:

Lines Parallel to the Coordinate Axes

A line is called a parallel line to the x-axis if the equation is in the form of y = a, where a is the distance of the line from the x-axis and a is a constant value. Similarly, A line is called a parallel line to the y-axis if the equation is x = a, where a is the distance of the line from the y-axis and a is a constant value.

Linear Equations in Two Variables: Previous Year Question and Answer

Question 1:
For what value of m will the system of equations 18x – 72y + 13 = 0 and 7x – my – 17 = 0 have no solution?

Solution:
To have no solution, the two equations should represent parallel lines.
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, to be parallel lines $\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$ this formula must satisfy.
18x – 72y + 13 = 0---(1)
7x – my – 17 = 0---(2)
So, $\frac{18}{7}=\frac{-72}{-m}\neq\frac{13}{-17}$
$⇒18m = 72×7$
$\therefore m =28$
Hence, the correct answer is 28.

Question 2:
If two LED TVs and one mobile phone cost INR 31,000, while two mobile phones and one LED TV cost INR 35,000, then the value of one mobile phone is:

Solution:
Let $x$ be the cost of an LED TV and $y$ be the cost of a mobile phone.
$2x + y = 31000$ ------------(i)
$x + 2y = 35000$
⇒ $x = 35000 – 2y$ ------------(ii)
Substituting (ii) in (i),
$2(35000 - 2y) + y = 31000$
⇒ $70000 - 4y + y = 31000$
⇒ $3y = 70000 -31000 = 39000$
⇒ $y = 13000$
Cost of a mobile phone = INR 13000
Hence, the correct answer is INR 13000.

Question 3:
If A is greater than B by 7, B is greater than C by 16, and A + B + C is 255, then the value of 3A + C – 4B is:

Solution:
According to the question,
A = B + 7
B = C + 16
Then, C = B – 16
Putting the value of A and C in the given equation
A + B + C = 255
⇒ (B + 7) + B + (B – 16) = 255
⇒ 3B – 9 = 255
⇒ 3B = 255 + 9
⇒ 3B = 264
⇒ B = $\frac{264}{3}$ = 88
Since A = B + 7 = 88 + 7 = 95
And C = B – 16 = 88 – 16 = 72
Now value of 3A + C – 4B
= 3(95) + 72 – 4(88)
= 285 + 72 – 352
= 357 – 352
= 5
Hence, the correct answer is 5.

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