Linear Functions in Grade V

Hi my dear friends! Once again I am here as math problem solver to solve you queries. In this article we will focus on the algebra topic of grade V. In the previous articles you learn relationship between data, Prime and composite numbers and variables, expression and graphs. Now, we move to the next topic of the same Grade in Algebra section that is how to use equations to solve several mathematical problems and graphing linear functions.  Before we move to, how to use equation to solve problems first focus on equations and its different types.  Equation is simply a combination of one or more terms that are joined by different operators and one most important operator that defines equation is called as equal to (=) symbol.  Equation may contain numerical, alphanumerical, expression, etc.  Equation is the simple form of representing mathematical statement and they are joined by an equal to sign.

Now, after brief introduction of equations let’s switch to the different forms of equations:-

1. The very first equation is that which have solution. In these types of equations we can find the value of different  variables, which are present in equations  like  x, y, z. let’s have its example, 

5a + b = 12, ….(1)

7a – 2b = 21…..(2)

As in these equations we are having two variables so, two equations are required in order to calculate the value of the variables. To solve such type of equations we can easily use the elimination or substitution method. 

Like in the above equation we can use elimination method, but for this we have to make the either value of ‘a’ or ‘b’ same in both the equations. in order to take the value of any of the variable same in both the equations simply multiply the equation one with 2 so that we can get the value of b same in both the equations. 

=>(5a + b) x 2 = 12 x 2, ………(3)

 =>7a – 2b = 21  …………(4),

10a + 2b = 24, 

7a – 2b = 21, 

Now, both the terms of’ b’ are same and having opposite side so we can simply add both the equations. On doing so, we get,  

13a = 45, 

So, a = 45/13, 

If we put the value of a in any of the above equation, we get the solution of the problem or we can say we get value of both the unknowns, which are commonly called as variables.  Put value of a in equation one and we get value of b as:

5 x (45/13) – b = 12, 

-b = (12 x 13) / (5 x 45), 

-b = 156 / 225, 

b = -156 /225. 

As I mentioned here that equation with solution, need some method to get the solution of several problems. we can use the following methods in order to get the solution of equations:-

a) Elimination method.  In this we find the solution of the equation in such a way that we try to eliminate any of the variable and by finding the value of one we replace its value and find the value of the other variables. 

b) Substitution methods: replace the value of the variable with some numeric value.  

c) Graphing method in this we represent the equation on graph and get the solution on graph. 

d) Trial and error method: in this method we find the solution of problems by fixing things, or by obtaining knowledge

e) Taylor method: A Taylor series method is of general applicability and it is the standard to which we can compare the accuracy of the various numerical methods for solving an I.V.P. 

f) Numerical method:  in this we do study of algorithms that use numerical approximation for the solution of the problems of mathematical analysis.

g) Solving equations using inverse functions: in this we use to solve an equation in order to find what values fulfill the condition stated in the form of an equation.

Generally we use the above two methods to solve the problem. In Grade V you learn only these two methods of solving the equation. The other methods are used to solve complex equations and you learn these methods in senior grades.

Now, move to the next type of equation i.e. equation without solution these type of equation have no solution that means we cannot determine the value of the variables. 

Let’s see an example of this, that what we mean by no solution problems, 

0 = -2, 0 = 12, 7=8

Examples: 5x + 4 -7x = 6x +7 – 8x – 5

Step:1. 4 + 5x – 7x = 6x – 8x + 7 – 5

Step:2. 4 – 2x = -2x + 2

Step:3. 2x – 2x = 2 – 4

Step: 4. 0 = -2. 

The other type of equation includes: 

Linear equation, Radical equations, constant equations, quadratic equation, equations with different variables and exponential equations. Let’s have a gist of all the different types of equations, start with the very first type of equations, 

a) Linear Equation: this is the most common type of equation which plays a vital role in mathematics. A linear equation is similar to an algebraic equation. The linear equations standard form is given as: 

y = mx +c, 

In this equation, m is the slope of the line and b is defined as the y intercept. Example of linear equation are: x= 3y + 15, 

X + y = 20, and other such type of examples. 

b) Radical equations: This type of equations defines the fractional exponent over variables. Fractional exponent is one of the ways to represent the radical terms or roots.   

Example: y + root 14 = 67, 

Root x + 5 = -9, 

c) Constant equation:  In the constant type of equation variable value cannot be changed. For example: x + 5 = 0,  2a + 6 = 7 . 

d) Quadratic equation: An equation which possess one or more terms but the higher degree of the equation is two. The degree is neither less than two not greater than that. The standard form of the quadratic equation is given as: 

ax² + bx + c =0. Let’s see few example of the quadratic equations,

1) 4a² + 6b - 89 = 0.
2) 4z² + 44z + 14 =0.
3) 5b² – 5b = 16.

e) equation with different types of different type of variables,

 1) Equation with one type of variables: The equations which possess only one variable is known as one type variable equation. Such as:

1.) 11x + 9 = 14

2.) 5p - 4 = 0

3) 6a = 6

2) equation with two different types of variables: in this equation we have two variable like x and y., a and b, etc. example of two types of variable: 

7a + 76b = 12

2.) 8p – 8q = 74

3) 34a + 36b - 72 = 0.

3) Equation with three kinds of variables: The equation which has only three types of variable is known as equation with three types of variables. Like a, b and c are present in the same equation or other such type of variables may also be present.

Examples: a.) 52x + 52y - 22z = 72

a.) 13a - 62b + 23c = 34

c) 1p + 22q -72r + 12 = 0.

f) Exponential equations: This type of equation is somewhat similar to exponents and they contain the powers in the equations. here are few examples of such type of equations that includes: 

a^b Here "a" is base and "b" is exponent.

Here, "a" multiplies "b" times.

2.) 4² Here 4 is base and 2 is exponent.

Here, 4 multiply 2 times.

(4)(4) = 16.

g) Differential equations: you learn this type of equation in higher classes. 

After this, now move to the other topic of today that we will study. After the equations to solve problems let's switch gear towards the Linear functions. What do you mean by the term linear equation? A polynomial function of single degree is called as the linear functions.  In this, we related dependent variables with another independent variable in a simple manner. If we have any mathematical equation in which there is no independent variable that is raised to a power which is greater than one, then we will get a straight line if we trace a graph of the simple linear function with one independent variable like y = mx + c. and you all know that this is the standard form of linear equation so it must give a straight line graph. 

Let’s move on the different form of the linear functions. To define linear function f with the first degree equation: f = ( X, Y)/ Y = mx + b where m and b are constants, x and y is called a linear functions. The function derives a straight line while graphing.

Mainly, three type of linear functions are defined and they are  given as:

1. Slope-Intercept Form 

 y = mx +b.

2. Point Slope Form 

m = (y - y1) / ( x – x1).

3. General Form is given as:

 Ax + By + C = 0.

Do practice of the entire concept daily and become the master of the subject.

In upcoming posts we will discuss about Patterns and Relationships in Grade V. Visit our website for information on 12th biology syllabus Maharashtra board