Linear Equations VII Grade

Hello my dear students, today we will focus on few more topics of algebra that you study in class VII. As you all now algebra is an interesting subject if you study it with interest and one of the boring subject if things go out of your mind then it is one of the difficult subject. In this article we will talk about Linear/non-linear functions, equations, inequalities. What do you understand by linear function? A linear function can be defined as a polynomial function of single degree. In this we relate a dependent variable with an independent variable in a simple way. It is that mathematical equation in which there is no independent variable that is raised to power greater than one. We can define the simple linear function as a linear equation which is having a single independent variable (y = a + bx), which results in a straight line when we plot the same on the graph.  

Now, let’s switch gear towards the form of linear function. In order to define the linear function we need an equation of first degree. The general form of linear function of first degree is given as:

f= (x, y)/ y = mx + b

In this m and b are constants, x and y are the linear function. When we plot the graph of same it will give a straight line. There are three main forms or types in linear functions.

1. Slope- intercept form: its general form is given as: y = mx + b,

 

2) Point slope for, its general form is given by m = (y –y1) / (x – x1)

3) General Form is given as: Ax + By  + C = 0.

Now, have a look on how to solve the linear functions. Solving the given function by altering its position and simplifying the equation to get Y is generally known as solving linear equations. The solution of these equations provides solutions of the corresponding practical problems. Here are few steps that you can use when you deal with the linear function in word problem form:

1. Read the given problem properly and then note down what is given to us in problem and what other things are required.
2. Denote the unknown quantities by literals like x, y, z, u, v, etc.
3. Translate the statements of the problem step by step in mathematical form so that we can get what to do.
4. After that, look for the quantities that are equal. And then make the equations corresponding to these equality relations.
5. Use an appropriate method and solve the equation formed in step 4.
6. It is the last step and in this you have to check the solution of the problem by substituting the value of the unknown found in step 5. Now, let’s have few examples of linear functions so that you can easily understand the concept and process of solving linear function problems.  

Example 1: Solve for a and b, where, b = a+ 13 and 2a ? b ? 10 = 0

Solution: in this we can easily put the value of the b as b = a + 13 in the equation

      2a ? b ? 10 = 0  

 2a ? (a + 13) ? 10 = 0,

a ? 23 = 0,

a = 23.

Now put the value of a  23 in 2a ? b ? 10 = ,

? y + 36 = 0,

? y = ? 36,

y = 36.

After linear function it’s the time to move towards the next topic that is called as the non- linear function. In simple terms we can define non-linear function as an equation whose graph is not linear and an equation with degree two or more than two. The general form of the same is given as:

f(x) = +….+ a₁x¹ + a₀ where a₀, a₁ ..a_n are stables. In the non linear function a_n is defined as a primary co-efficient and a_n x_n is principal term. The greatest degree of non-linear function is greater than two or similar to two. In non linear function, graph can be a curve, zig- zag line, or any even shape but it can’t be a straight line. The Non linear function can be defined in quadratic form, exponential form and logarithmic form.

Different types of non linear function deals with solving several types of polynomial function where as polynomial function is also called as non linear function.  We can solve the non linear function using substitution method or quadratic equation operations, etc. 

 

Non linear functions play an important role in algebra. Any of the function is not a linear function and can’t be a complete linear function by transforming the Y variable.  

Normally, there are three types of non linear function that we use in mathematics, they are given as:

1. Exponential function
2. Quadratic function
3. Logarithmic function

This is the time to see few examples of the non linear functions.

Example 1: x² - x – 12 = 0, calculate the value for ‘x’ for the given non linear function.

Answer:

As you all know the given equation is in form of x² – 4x +3x - 12 = 0, which resembles with the general form of quadratic equation.

Now, we can get the value for x from the primary term and 3 from secondary term.

x (x – 4) + 3(x - 4) = 0

Now combine the like term (x -4)

(x + 3) (x - 4) = 0

To get the value for x we can associate the factor to zero

a + 3 = 0 or a – 4 = 0

a = -3 or a = 4

a = 4

Thus, the factors a₁ and a₂ are -3, 4.

Example 2:

p² - 3p – 10 = 0, determine the value for p for the non linear function.

Answer:

Now, we can find the factor for the given quadratic equation

p² – 5p +2p - 10 = 0

Now, get the value for x from the primary term and 3 from secondary term.

p(p – 5) + 2(p - 5) = 0

Now we can combine the similar term (p - 5)

(p + 2) (p - 5) = 0

To get the value for x we can associate the factor to zero

p + 2 = 0 or p – 5 = 0

p = -2 or p = 5

Thus, the factors p₁ and p₂ are -2, 5.

Example 3:

x² - 6x + 5 = 0, get  the value of x for the given non linear functions

Solution:

First we can learn the factors for the given quadratic equation

Sum of the roots (-6) = (-5) + (– 1)

Product of roots (5) = (-5) × (-1)

By combining these two roots we can obtain the factor form as,

x² – 5x - x + 5 = 0

Now get the value for x from the primary term and 5 from secondary term.

x (x – 5) - 1(x - 5) = 0

Now we can combine the similar term (x - 5)

(x - 5) (x - 1) = 0

To get the value for x we can associate the factor to zero

x - 5 = 0 or x – 1 = 0

x = 5 or x = 1

Thus, the factors are x1 and x2 and the values are 5, 1.

Now, switch to other topics Equations and inequalities. You all are familiar with both these terms. Here, I will give a brief introduction of both the topics as you are reading the same from earlier classes.  Equations are those mathematical statements that are joined together with help of equal to symbol. There are many different types of equations like linear equations, quadratic equations, polynomial equations, and many other types of equations. There are several different types of methods that you can use to solve different types of equations like linear equations can be solved by simple motion of the variables and constants.

Inequalities can be defined as an expression which is defined using special symbols.  Mainly four different inequality symbols are used using which inequalities are defined like: greater than (>), greater than equal to (<), less than (>=), less than equal to (<=). Solving linear equalities is very simple and you can easily solve them. Solving linear equalities are very similar to solving a linear equation. Inequalities give infinite number of solutions and the answer is right only when the inequality is true.

In all the above mentioned topics for any problem you can take online help and solve all your problems.