System of Equations and Inequalities of Ninth Grade

Friends today we all are going to understand about the basic concept behind one of the most interesting and a bit complex topic of Ninth Grade mathematics that is system of equations. Till eigth grade we had equations but they were easier to solve but as we are moving towards the ninth grade, equations become little complex. We all now have the basic knowledge of equations as used in mathematics, but let us do a quick recap. In simple mathematical manner we can say that an equation consist of two algebraic expressions and 'is-equals-to' symbol (=) is used between the two expressions. An algebraic expression is one which consists of variables and constants and if the equation is true for all the variables used then it is called as identity. For example

X + Y = Y + X (an example of Identity)

Now the next question that comes to our mind is, How to solve an equation? Lets take an example to understand it better: an equation 4x + 7 = 31. now we have to figure out the value of variable x where this equation holds true and the procedure of figuring out this unique value of x is what we called as solving an equation.

4X + 7 = 31 can be written as 4x = 31 – 7 = 24

now x = 24/4 = 6.

We already know that a linear equation is the one in which we always get a straight line while graphing it. Also the linear equation in one variable is an equation of the form :

ax = b where a and b are the constants and x is the variable whose value need to be determined.

Now I am going to tell you about system of equation of grade nine or the ninth grade that is the next step after the linear equations. but before we move on we should understand what is a system of equations: A systems of equations come with the relationship between two or more functions, which can be used to form a number of real-world situations. It is basically a collection of two or more equations with a same set of unknowns. An equation can be linear or Non linear. In the ninth grade we have to learn about system of linear equations. A system of Linear equations is a collection of linear equations that come with the same set of variables.

An example will help us in understanding it better, suppose there are two variables “a” and “b” and we have two options.

a + b = 5

2a – b = 1

Lets starts it by finding out whether both the equations are satisfied if a = 2 and b = 3. This is easy to check as I know how I can get the result for the equations, but how we are going to arrive the answer if nobody gives us the solution? So, the real issue is how to find out the exact solution for a system of equations.

In general our course books provide us two methods of solving a linear equation that are sibstitution and elimination method (also called as Gaussian Elimination). Substitution method works by solving on equation for one variable and substituting its value in other equations. This method is basically used for small systems and it can be applied in nonlinear systems to get the desired results. Elimination or the Gaussian Elimination method becomes increasingly important while solving large linear system of equations.
An example will definitely help in a better understanding, lets continue with the same example. Multiply the first equation with 2 in both the sides. Now the equation becomes 2a + 2b = 10 and the another equation becomes standard or unchanged as : the two equations are 2a + 2b = 10
2a – b = 1
As in the above equations the variable “a” has the same coefficient in both equations. Now the next step is to subtract 10 from both the sides of second equation. It looks like
2a – b – 10 = 1 – 10
Now, as we already know that 2a+2b = 10 so, lets substitute number 10 with this value :
2a – b - (2a + 2b) = -9.
Now we need to simplify the above equation in following manner :
2a – b – 2a – 2b = -9
So we get – b – 2b = -9
And further simplifying it we get : - 3b = -9
Divide -3 from both the sides of the equation, we get
b = 3
Now substitute the value of b in the first equation to get the value of another variable :
2a + 2(3) = 10
2a + 6 = 10
2a = 4
a = 2.
In the end we can check our result by substituting the values of “a” and “b” in the equations. Clearly substituting a = 2 and b = 3 in the equations if the values present true answer for the equation then the result is correct (means LHS should be equivalent to the RHS).

Linear systems can be represented in matrix form as the matrix equation: Ax = b. Here A is the matrix of coefficients, x is the column vector of variables, and b is the column vector of solutions. An example to show a system of equations:

2x + 3y – z = 1

3x – 4y – 7z = 8

-x + ¾ y – 2z = 4

It is a system of three equations in the three variables x, y, z.

Next lets think of a system of equations that has more than two equations, say “n” equations, to solve such kind of problem we need to pick one of these equations, it could be the first one or any in the system of equations and a particular variable. Subtract suitable multiples of our selected equation from the other equations so that the selected variable no longer remains in the resulting n – 1 equations with n – 1 unknowns. Now in next step we need to repeat the same process until we get a single equation in a single unknown (means we need to further simplify the equation to get the simplest form of single equation). This whole process is known as Gaussian Elimination.

The next important topic to be discussed is Inequality. In 8^th and 9^th standard inequality problems are very easy to solve. As in ninth grade the syllabus has several problems involving linear inequalities. Before we go on to solving these problems, we should understand the inequality. An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. Slope formula plays an important role in graphing linear inequalities. So we need to know is this slope formula means and how to use it. Slope of a line describes the steepness, incline or grade of the straight line.

The slope through the points (x1, y1) and (x2, y2) is given as Slope formula.

( m = y2-y1/x2-x1 ) where x1 is not equal to x2.

In general slope intercept form denotes the formula : y = mx + b.

Where m = slope of the line

b = y intercept.

If we what to understand it in detail lets start with basic information that an inequality consists of two algebraic expressions combined by one of the inequality symbols like < means less than and the symbol > means greater than and the symbol ≦ or ≤ less than or equal to etc. There symbol ≠ is also used, but the other cases occur more frequently.

A linear inequality describes an area of the coordinate plane that has a boundary line. In simple words a linear inequality represents everything on one side of a line on a graph.

In mathematics a linear inequality is an inequality which involves a linear function

for better understanding here is an example -4 < 1

If we multiply -1 from both the sides we get the values 4 and -1, but

4 > -1.

The multiplication with negative value reverses the inequality sign. So generally when we solve problems problems involving inequalities the procedure is very similar to the way we the equality problems. But one thing that we have to always be careful about is to reverse an inequality sign when we multiply on both sides with the same negative value or factor.

Here is one more ninth grade example to understand it better :

3a + 4 < 5

To solve such type of inequality we need to subtract 4 in both the side :

we get : 3a < 1

now further simplifying it we need to subtract both the sides by 3

a < 1/3 is the required solution.

one more example to understand the basic concept behind reversing the symbols. This example shows you that how to reverse a symbol or sign when negative value is used in the equation.

Equation is : 3a + 4 < 5a – 2

now what we have to do is to subtract 4 and 5a from both the sides

we get – 2a < -6

then in next step we simply needs to divide both the sides by - 2 and just reverse the sign to obtain :

we get a > 3

But once you get the solution always check it, for this  we just need to replace the inequality sign with an equality sign and just substitute the value of our final result in the equation. If the result represent true value for the equation then your result is right else you will have to do it once again.