Algebraic Equations: Concepts & Theory

Just assume that I am carrying some amount of rupees. When I make it thrice of what I am carrying and add Rs. 150 given by my mom, then the total amount I am carrying now is Rs. 1050. Could you tell me what was the initial amount that I was carrying?
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This is where the concept of EQUATIONS comes into picture. What we need to do in this problem is, we will make one equation in one variable and get the value of that variable which would be our answer. But before making that equation, we come across some new terms which need to be understood first. The first new term used is EQUATION: What is the meaning of this word EQUATION? An equation is basically defined as the relationship between variables with equality sign. Next term being introduced is variable which means anything which is not constant or is varying all the time is known as variable. Let us now try to solve the above question with the help of an equation.
Supposing the amount I was carrying initially was Rs. x. As I made it thrice of that, it means now it has become 3x. Also I added Rs. 150 given by my mom to this, so new total becomes 3x + 150. This value is given to us as Rs. 1050. So we can form the equation as 3x + 150 = 1050. Solving this equation, we get x = Rs. 300.
This was a problem dealing with equation in just one variable. These types of equation are called linear equations. So we can have linear equation in one variable or two variables or three variables.
We also define another term called as degree of an equation. By degree of an equation, we mean the highest power of the variable. Thus we confirm that degree of linear equation is 1. Let me take you to a particular section of 10th class of 80 students (boys and girls) where they were planning to throw a party with Rs. 10000 collected with each boy contributing Rs. 100 and each girl contributing Rs. 150. Could you tell me how many boys and girls were there in the class?
This is also a case of linear equation but not in one variable, it is in two variables because we are dealing with two variables (boys and girls) in this problem. Now let us understand the problem and try to solve it:
Let us take number of boys as x and number of girls as y. Since there are total 80 students, so we can say that x + y = 80. Also it is given that each boy contributes Rs. 100 and each girl contributes Rs. 150 and total contribution is Rs. 10000, so we can another equation as 100x + 150y = 10000. Solving linear equations, we get the final answer as x = y = 40.
Now let us consider a system of two linear equations:- M1x + N1y = q1 and M2x + N2y = Q2. This pair of linear equations has:-
  1. A unique solution if, (M1 / M2)≠ (N1 / N2) = Q1 / Q2
  2. Infinite solutions if, (M1 / M2) = (N1 / N2) = Q1 / Q2
  3. No solution if, (M1 / M2) = (N1 / N2) ≠ Q1 / Q2
In this article, we have covered concepts of equations and linear equations but we have not covered the concepts like differential equations. To be a good equation solver, you need to understand the approaches given in the solved examples.
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