Algebraic Equations: Overview, Questions, Preparation

An algebraic equation is a mathematical statement in which two statements are set equal to each other. Here a variable is defined by an alphabet. There may usually be a single value of the variable or multiple values for which the mathematical statement is true.
The two expressions are related to each other by an equal sign.

A fundamental example of the equation is:
1+7= 8
Here, the left-hand side expression is equivalent to the expression on the right, which satisfies all the necessary conditions for it to be considered an algebraic equation.

Types of Algebraic Equations

The types of algebraic equations are:

i) Polynomial equations: It is an equation in which only one variable is being used in any of the two expressions. Sometimes an equation has more than one variable; such an equation is called a polynomial equation. 

Example, 2a+4b=7

ii) Quadratic equation: A quadratic equation is an equation with a degree 2 or one of its variables having value two, or the highest exponential value in one of its variables is a square or to the power 2. In a graphical form, it resembles a parabola.

Example, 2a² +7b-4=70

iii) Cubic equation: It is an equation with the degree as 3. It represents a cubic curve that has three values for a single variable.

Example,7b³- 4a= 6

iv) Rational equation: This is an equation in the form of a rational identity where the numerator is an equation, and the denominator is a non-zero value equation. 

(2a+7b)/7=21b

v) Trigonometric equation: It is an equation based on trigonometric functions like sin, cos, tan, and so on. 
Its graph resembles the trigonometric function, which is being used. 

Example, 2asinӨ = 45

Students are made familiar with this concept in class 7. Students will study this topic up to class 10, and it has a weightage of 20 marks in the final board paper. In the 12th boards, the mark weightage for algebra is 10 marks. But it is the foundational concept for most of the other chapters. 

1: Solve the following equations:
 2(x + 4) = 12

Solution:

=> (2(x + 4))/2 = 12/2
=> x + 4 = 6
=> x = 6 – 4
=> x = 2

2:. A rectangular plot is 456m². The length of the plot (in metres) is one more than thrice its breadth. Find the length and breadth of the plot.

Solution:

Let the breadth of the rectangular plot = q m 

Then, length of the plot = (3q + 1)m

Area of a rectangular plot = 1qb.

=> 456 (3q + 1)q
=> 456 = 2q^2 + q
=>3q^2 + q- 456 = 0 is the required quadratic equation

3. Keshav scored twice as many runs as Vishal. Together, their runs fell two short of a double century. How many runs did each one score?

Solution:

Let us assume Vishal’s score is x
Then, Keshav’s score is 2x

According to the problem, their combined score is 2 less than a double century. 
=> Vishal’s score + Keshav’s score = 200 – 2
=> x + 2x = 198
=> 3x = 198
=> 3x/3 = 198/3
=>x = 66

So, Vishal’s score is 66.

And Keshav’s score is 2x = 2 × 66 = 132.

Q: What is an equation?

Q: What is an inequality equation?

Q: What is the general format of a quadratic equation?

Q: Can a variable have more than one value?

Q: Can a variable have no value?