Hello my dear friends today we will learn few interesting topic of grade X that are imbibe in Algebra. Algebra is a very large subject that has lots of topics and you can easily learn them with intuition. Mathematics is that subject that can only be learnt if you have intrusion and is considered as one of the most interesting and important subject. Many students are math phobic and they always try to move away from this subject. Friends no need to get afraid of this wonderful subject you can easily learn this subject with help of you teachers, tutors and me. In this article we will focus on changing parameters of function, and system of equations. Before this in the previous sections I made you familiar with other math topics like, Different forms of numbers, Scientific notation, square roots, exponents, radicals, absolute value, factorial, logarithms., Properties of numbers, Estimation of solutions, Sequences and series Proportionality/direct variation, and exponential/fractional expressions.
Now, start talking about our today’s topic i.e. changing parameters of function. First talk about the parameters, in common meaning, the parameters are used to identify a characteristic, a feature, a measurable factor, which can be used to define a particular or specific system. For the evaluation or for the comprehension of an event, or a project or any other such type of situation, parameter is an important element. If we talk about parameters in Mathematics then a parameter is a defined as a quantity that serves to relate different functions and different variables using a common variable when such a relationship would be difficult to explicate with an equation.
Different Mathematical functions have one or more arguments that are designated in the definition by variables, on the other side their definition also have parameters. In the list of arguments variables are mentioned that the functions takes, but the parameters are not mentioned. When we define parameters or any parameters present, the definition actually defines a whole family of functions, one for every valid set of parameters value. We can take the example of the general quadratic equation form: that defines the function and parameters.
F(x) = ax2 +bx +c,
In this equation, variable x designates the function argument, but a,b, and c are parameters that determine which quadratic function one is considering. The Parameters could be incorporated into the function name to indicate its dependence on the parameter. To understand this we can take example of log, such as:
loga(x) = log (x)/ log (a),
In this ‘a’ is a parameter that is indicating which logarithmic function is being used. Here ‘a’ is not an argument of the function.
Parameters varies from function to function, in Grade X, you commonly use population parameter. A quantity or statistical measure that for a given population which is fixed and that is used as the value for a variable in some of the general distribution or frequency function to make it is a descriptive of that population.The variance and the Mean of a population are referred as the population parameter.
The next topic that we will discuss today is called as system of equations. A system of equation means a set or a group of equations and sometimes we refer this as simultaneous equations also. In this we have more than one equation with multiple variables and we try to solve the equation in order to calculate the value of the all variables. Linear equations are very simple if we compare them with non-linear equation and the simplest linear system is one with two equations and three unknowns or variables. In simple words we can define the same as a collection of two or more equations with a same set of unknowns. For the solution of system of equations, we need to find values for every of the unknowns that will declare every equation in the system. When you deal with system of equations you can either have linear equation or non-linear.
Now, let’s focus on the types of systems of equations. In grade X you will learn Elimination, Substitution, linear equation, matrix, and consistent equations. Now, have an introduction of all the topics that I mentioned here. First talk about the elimination method or elimination technique, it is considered as one of the algebraic method that we use for solving the system of equations. In elimination method we perform operation on the one equation the operation may be of adding some number or multiplication, etc. we perform this operation in order to cancel one of the variable and with this we can easily find the value of other variables.
The next one is Substitution method, this method the algebraic expression of one of the variable is substituted in another equation at the place of the respective variable and then the variable is used to solve, when we solve the variable we get the numeric value and then easily with its help we can solve the different problems. by substituting in any of the equation the value of the second variable is also found easily. The third one is Linear equation, this is also considered as an algebraic expression, which relates the two different types of variables and produces a graph which is always in the form of a line.
Fourth one is Matrix, it is a rectangular array of numbers and in this we use to write numbers inside the bracket. Matrix method is used to find the solutions for complex systems of equations. The last one is called as the consistent System. In this we have set of equations whole solution set is represented by only one ordered pair. Here are few examples for you students so that you can easily understand how to solve the system of equation. Examples are the best way to understand different type of problems and with this you can solve multiple problems in an easy way.
Here, is the very first and simple example, give the solution of the problems and tell whether it is example of system of equation or not.
y + 21 = 71
Solution:
This is a simple example of linear equation as only one equation is given but in system of equation we have more than one equation that either is two, three or more. We need not to implement any of the above mentioned method in this as we can easily solve the problem by subtracting 21 from both the side of the equation.
Y + 21 – 21 = 71 – 21,
Y =50,
In this way we can easily calculate the value of the unknown in this equation.
Now, here is another example for you. Give the solution of the problems and tell it is system of equation or not.
Example: b = 2a + 1, 2b = 3a – 2
Solution: this is a system of equation as it is having two equations, and for solving this problem you can easily use any of the above mentioned method which we use to solve the system of equations. Let’s use the substitution method and find the solution of this problem.
Step 1: substitute the one equation to another equation like:
2(2a + 1) = 3a – 2,
Step 2: now, we have single variable equation, we know how to solve that variable equation as the equation is now in form of single variable which you can easily solve and find its value.
4a + 2 = 3a – 2,
4a – 3a = -2 – 2,
a = -4,
In this way we find the value of the one variable now you can easily find the value of the other variable by putting the value of ‘a’.
Step 3: to find the value of b put value of a
b = 2a + 1,
b = 2(-4) + 1,
b = -8 + 1,
b = -7,
Thus we find the value of ‘a’ and ‘b’ using substitution method.
One more example of the system of equations and solve this problem using the elimination method.
Example: 2p + 2q = 4
4p – 2q = 8.
Solution: As there is two equation given to us, so we can easily say that it is a system of equation having two variables p and q.
Step 1:
2p + 2q = 4.
4p – 2q = 8.
Step 2: in this subtract equation 2 from equation 1 and on doing this we get,
6p = 12
Step 3: now, divide both the side by 6, on doing this we get
p = 2.
Step 4: now, replace p with its value in the equation 1, when we do this we easily calculate the value of the other variable q.
4p – 2q = 8.
4(2) – 2q = 8.
8 – 2q = 8.
q= 0.
Now, we have both the value of p and q
(p, q) = (2. 0).
In other method you will study in the higher classes.
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