Notations
Algebraic Notations | Meaning | Example |
x = y | x is equal to y | 1 = 1 |
x â‰ y | x is not equal to y | 1 + 1 â‰ 1 |
x < y | x is less than y, or, y is greater than x | 2 < 5 |
x > y | x is greater than y, or, y is less than x | 4 > 3 |
x â‰¤ y | y is greater than or equal to x, or, x is less than or equal to y | |
x â‰¥ y | x is greater than or equal to y, or, y is less than or equal to x | |
x â‰ˆ y | x is approximately equal to y | 2.99999 â‰ˆ 3 |
Additions/Subtractions
Statement 1 âˆ’ Ram is having 5 more apples than Shyam.
â‡’ Number of apples with Ram = Number of apples with Shyam + 5
Assume that number of apples with Shyam = x, then number of apples with Ram = x + 5
Alternatively, if we assume that the number of apples with Ram = y, then number of apples with Shyam = y âˆ’ 5
Multiplication
When there are two or more equal numbers to be added together, the expression of their sum may be abridged.
For example,
x + x = 2 Ã— x = 2x
x + x + x = 3 Ã— x = 3x
x + x + x + x = 4 Ã— x = 4x
In this manner, we may form an idea of multiplication; and it is to be observed that,
x + x = 2 Ã— x = 2x
x + x + x = 3 Ã— x = 3x
x + x + x + x = 4 Ã— x = 4x
2 Ã— x signifies 2 times x or twice x
3 Ã— x signifies 3 times x or thrice x
Further, we can multiply such products again by other numbers; for example:
2 y Ã— 5 = 10 y
2 y Ã— 5 z = 10 yz
2 y Ã— 6 y = 12 y^{2}
Linear Equation
An algebraic equation, such as y = 2x + 7 or 3x + 2y âˆ’ z = 4, in which the highest degree term in the variable or variables is of the first degree. The graph of such an equation is a straight line if there are two variables.The general form of linear equation in two variables x and y is ax + by + c = 0, a â‰ 0, b â‰ 0, and a, b and c real numbers. Here a and b are known as co-efficient of x and y respectively and c is a constant. A solution of such an equation is a pair of values, one for x and the other for y, which makes LHS and RHS of the equation equal.
For a linear equation in two variables x and y:
ax + by + c = 0
In the above equation, for every real x, there exists a real number y corresponding to x.
Therefore every linear equation in two variables has infinitely many solutions, i.e., infinitely many pairs (x, y).
For example, the equation 2x + 3y = 10 will have infinite solution.
All these solutions are represented by points on a certain line. Due to this fact only this equation is called LINEAR because the graph of the equation on the x-y Cartesian plane is a straight line.
For example, consider the equation
All these solutions are represented by points on a certain line. Due to this fact only this equation is called LINEAR because the graph of the equation on the x-y Cartesian plane is a straight line.
....(1)
To find a solution of linear equation in two variables, we assign any value of one of the two variables and determine the value of the other variable, from the given equation (1).
Thus, taking x = 1, we get corresponding value of y =
Similarly, taking x = 0, we get y = 4; and so on.
The following table lists six possible values for x and the corresponding values for y, i.e., six solutions of the equation:
To find a solution of linear equation in two variables, we assign any value of one of the two variables and determine the value of the other variable, from the given equation (1).
Taking any two pairs of given equation, we plot corresponding points, say, P and Q. The line PQ through these points is related with the given equation in the following manner:
- Every solution x = p and y = q of the given equation determines a point (p, q) that lies on this line.
- Every point (x_{i}, y_{i}) lying on the line PQ, determines a solution x = x_{p} y = y_{i} of the given equation.
The line PQ is said to be the graph of the given equation. It is worth noting that:
- We can add or subtract any number on both sides of the equation without affecting the equation and its solution.
- We can multiply or divide both sides of an equation by a non-zero number without affecting the equation and its solution.
If we plot the solutions of the equation 3x + 2y = 8, which is represented in the table above then we notice that they all lie on the same line. We call this line the graph of the equation since it corresponds precisely to the solution set of the equation.