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Linear Equations and Inequalities









Introduction:

                  When we come across real situations like " If the incomes of two persons is in the ratio 3 : 6 and their expenditures are in the ratio 2 : 5, find their income  if both of them save 400 dollars.". It is very difficult to slove this by trial and error method.  These types of problems can be solved easily using linear equations. Let us study about the methods of solving equations and inequalities.

Linear Equations and Inequalities in one variable:

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               General form Linear Equations of of one variable is ax + b = 0, where a and b are real numbers and a not equal to 0. The only solution to this linear equation of one variable is x = - b/a.
Solution: The value of x which satisfy the equation is called the solution to the equation.The solution to the linear equation of one variable can be represented on the number line.

Example: solve 2x + 5 = 13, and represent the solution on the number line

Solution :   We have 2x + 5 = 13
=>                      2x + 5 - 5 = 13 - 5
=>                                2x = 8
=>                                  x = 8/2 = 4
graph11
Hence We can see the solution  being represented on the number line.
Linear Inequalities:
             
A statement of inequality between two expressions involving a single variable x with highest power 1, is called a linear inequality..
Examples : 2x + 4 < 5,    6(x - 4) > = 5x -2
Domain of the variable: The set from which the values of the variable x are replaced in an inequality, is called replacement set of the domain of the
variable.
Solution Set: The set of all values of x from the domain (replacement set) which satisfy , the given inequality  is called the solution set of the inequation.

Example 1 : A = { x : x belongs to N, X < 9 }

Solution: We have A = { x : x belongs to N, x < 9 }
                           A = { 1,2,3,4,5,6,7,8}

Example 2: B = { x : x is a positive integer, -4 < x <  0 }

Solution: We have B = {   } , since there is no positive integer in this interval.

                          

How to solve Linear Inequalities:

Rules of Inequalities:

Rule 1: Adding same number on both sides does not alter the inequality.
Example:             When 5 > 3 then 5 + 2 > 3 + 2 => 7 > 5
Rule 2: Subtracting the same number on both sides, does not alter the inequality.
Example :            When 5 > 3, then 5 - 2 > 3 - 2 => 3 > 1
Rule 3: Multiplying  or dividing both sides by a positive number does not alter the inequality.
Example :            When 6 > 4 then 6 x 3 > 4 x 3 => 18 > 12
         
                            When 6 > 4, then 6/2 > 4/2 => 3 > 2  
Rule 4 : Multiplying or dividing both sides by a negative number will reverse  the inequality.
Example :            When 6 > 4, then 6 ( -2) < 4 ( -2)
                                
                                             =>        - 12 < -8
                    
                            When 6 > 4, then 6/(-2) < 4 / ( -2)
                                       
                                             =>          -3 < -2
Rule 5: When we take the reciprocals on both sides of an inequality, then the inequality reverses.
Example :              When 1/4  < 1/2, then 4/1 > 2/1
                                             =>               4 > 2

Example : Solve 2x + 5 >= 13. Represent the solution on the number line, if x belongs to set of Real numbers.

Solution:              We have 2x + 5 > = 13 and the replacement set is "Real Numbers"..
                                          2x + 5 > = 13
                         =>         2x + 5 - 5 > = 13 - 5
                         =>                   2x > = 8
                         =>                     x > = 8/2
                         =>                     x > = 4

graph12
In the above number line, the darkened portion x > = 4, is the solution set for the inequality.

Linear Equations and Inequalities in two variables:

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Linear Equations: The general form of linear equations of two variables are ax + by + c = 0.
             Here the variables used are x and y. As the value of changes the value of x also changes the value of y also changes correspondingly.
             Since x  can take infinitely many values on the real number line, there will be corresponding real value for y.
Example : Find any two solutions of the linear equation of two variables 4x + y = 8 hence represent it graphically.
When 4x + y = 8 => y = 8 - 4x
Let us assume some values for x
when x = -2, y =  8 - 4 ( -2) = 8 + 8 = 16
Hence one of the solution of the above equation is ( -2, 16 )
When x = 0, y = 8 - 4 ( 0 ) = 8 - 0 = 8
The other solution is ( 0,8).
Hence by substituting different values for x, we can find the corresponding values of y.
The pair of values (x,y) are called the set of solutions to the given equation.
When we plot the points on the two-dimensional graph, we get a straight line. The line will divide the graph into two regions.
The graph will be as shown below.

graph13

Linear Inequalities in two variables :
                               The general form of linear inequalities of 2 variables will be of the form ax + by < c, ax + by < = c, ax + by > c, ax + by > = c.
We know that the graph of the equation ax + by = c, is a straight line which divides the xy-plane into two parts which are represented by ax + by >=c,
and ax + by < = c. These two parts are known as closed half spaces.
                              The region ax + by < c and ax + by > c are known as the open half spaces.
                              These half spaces are known as the solution sets of the corresponding inequalities.
In order to find the solution set of a linear equation in two variables, we follow the following steps.
Step 1: Convert the given equation, say ax + by > = c, into the equation ax + by = c, which represents a straight line in  2 dimensional plane.
Step 2: Find the x and y intercepts of the equation ax + by = c, and plot the points..
Step 3: Join the points in step 2. Incase of strict inequality (i.e) with < or > sign, join the points by dotted lines, else draw a thick line.
Step 4: Choose a point , if possible (0,0) not lying on it. Substitute the coordinates in the inequation. If the equation is satisfied, then shade the portion of the plane which contains the chosen point, otherwise shade the portion which does not contain the chosen point.
Step 5: The shaded region obtained in step 4, represents the desired solution set.
Example : Represent the region which satisfy the two inequalities  4x + y <= 8 , and 15 x + 7 y > 105 in two dimensional graph.
Solution: Let  4x + y = 8 ------------------------------(1) and 15 x + 7y = 105 ---------------------(2)
Let us find the x and y intercepts of each line.

            4x + y = 8
       x         
           y          
       0
           8
      2
          0
Since the inequality is 4x + y < = 8
(1) We draw thick line in the graph.
(2) when we plug in (0,0) , we get 4(0) + 0 < 8 => 0 < 8, which is a true statement.
    Hence we shade the region which does contain the origin.
 
         15 x + 7y = 105
        x          
          y           
        0
        15
        7
         0

Since the inequality is 15 x + 7y > 105,
(1) We draw the dotted line in the graph.
(2) By substituting (0,0) we get 15(0)+7(0)> 105 => 0 > 105,
             which is a false statement.
     Hence we shade the region which does not contain the origin.
The graphs of these two inequalities are shown below.

graph14


Functions and Linear Equations and Inequalities:

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Functions: Functions are represented as y = f(x), where x is an independent variable and y a dependent variable.

Example : y = f(x) = 2x+3, y = f(x) = 3x2 - 2x + 6.
As the value of x varies, the value of y also varies.
The slope - intercept form of the linear equation of two variables, y = mx + b, where 'm' is the slope and 'b' the y-intercept of the line.

The graph of the line is a straight line which can be extended indefinitely on both the direction.

The Domain of the function is the set of all real numbers,
                                 x = { x : x belongs to Real numbers }
The range of the function is also the set of all real numbers which depen on x.
                                 y = { y : y belongs to real numbers, y = f(x) }
Example : Express the linear equation 2x + y = 6 in the form y = mx + b.
Find the Domain and Range of the function, where 0 < x < 5 and x belongs to Natural numbers

Solution:              We have 2x + y = 6
Let us write it in the form              y = f(x).
                                          2x + y = 6
                                   =>           y = -2x + 6
The domain likes in the interval (0,5), where x belongs to Natural numbers.
                                 Therefore, Domain = { 1,2,3,4 }
To find the Range:
               Substituting x = 1, we get, y = -2(1)+6 = -2+6 = 4
               Substituting x = 2, we get, y = -2(2)+6 = -4 + 6 = 2
               Substituting x = 3, we get, y = -2 (3) + 6 = -6+ 6 = 0
               Substituting x = 4, we get, y = -2(4) + 6 = -8 + 6 = -2
                                  Therefore Range = { -2, 0, 2, 4 }

Note: In the above example, the domain is restricted. Without restriction the domain of a linear equation of two variables vary from - infinity to + infinity.
                           

Practice Questions:

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1.  Solve the equation for x. 4 ( x- 5) = 40.
2. Check if x = 4 is the solution of the equation 6x + 4 = 20
3. Find any four solutions of the equation 3x + y = 10.
4. Find the solution set of the inequality, 2x + 5 < = 15, where x belongs to Real numbers.
    Express your answer in interval form and also represent on the real number line.
5. Graph the equation 4x + 7y = 28, by finding the x and y intercepts.
6. Shade the region which satisfy the inequality 3x + y > = - 2.

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