Linear Equations and Inequalities

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 Rs. 400." It is very difficult to solve this by trial and error method.  It is more important now than ever before for everyone, specially teachers and students to aware of latest changes in the Syllabus and the best new methods, programs, and devices in the field right away. We focus on simple ways of learning to inspire students. Unfortunately, many of us have treated algebra as a complicated subject and skip to solve related problems because of different reasons. We have solved linear equations and inequalities in the way that will help you to understand concept easily.

Definition

What are Linear Equations?  An algebraic equations with degree one are known as linear equations.
Examples: 2x + 1 = 0, 10y = 7 and 1/3x - 1 = 0

Some Important Points:
For x, y variables; p,q coefficients and s is a constant value.
1. General form: Linear Equations of one variable is px + s = 0.
Linear Equations of two variables is px + qy + s = 0
2. The only solution to the linear equation of one variable is x = - s/p.
3. The value which satisfy the equation is called the solution.
4. The solution to the linear equation of one variable can be represented on the number line.
5. When we plot a graph for an equation with 2 variables that gives a straight line.

What are linear inequalities? A statement of inequality between two expressions involving a single variable with highest power 1, is called a linear inequality.
Examples : 2x + 4 < 5,  6(x - 4) $\geq$ 5x -2

Some Important Points:
1. Domain of the variable: The set from which the values of the variable are replaced in an inequality, is called replacement set of the domain of the variable.
2. Solution Set: The set of all values of x from the domain (replacement set) which satisfy, the given inequality is called the solution set.

Solving Linear Equations and Inequalities in One Variable

A linear equation in 1 variable, say x, is an equation that can be written in the form: ax+b = 0 (a and b are real number and a $\neq$ 0). The properties used in solving linear equations and linear inequalities are very similar, except the multiplication property. Check for linear equations properties and linear inequalities properties for more understanding.

Linear Equations and Inequalities in Two Variables

The general form of linear equations of two variables is 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 correspondingly.
Variable, x can take infinitely many values on the real number line, there will be corresponding real value for y.
For Example : Find any two solutions of the linear equation of two variables 4x + y = 8. Represent 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).

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:

Linear Inequalities in two variables
The general form of linear inequalities of 2 variables will be of the form ax + by < c, ax + by $\leq$ c, ax + by > c, ax + by $\geq$ 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 $\geq$ c,
and ax + by $\leq$ 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.

Functions and Linear Equations and Inequalities

Functions are represented as y = f(x), where x is an independent variable and y is a dependent variable. Slope - intercept form is one of the examples of linear equations.

Example : Linear equations: y = f(x) = 2x+3, y = f(x) = 3x$^2$ - 2x + 6.
Whereas Linear inequalities: y = f(x) > 2x + 3, y = f(x) $\geq$,  3x$^2$ - 2x + 6

Few Important facts:
1. The graph of the line is a straight line which can be extended indefinitely on both the direction.
3. The Domain of the function is the set of all real numbers,
x = { x : x belongs to Real numbers }
4. The range of the function is also the set of all real numbers which depend on x.
y = { y : y belongs to real numbers, y = f(x) }
5. A linear equation in one variable have one solution whereas a linear inequality can have many.

Examples

Example 1Solve 2x + 5 = 9, and represent the solution on the number line
Solution :   We have 2x + 5 = 9
=> 2x + 5 - 5 = 9 - 5
=> 2x = 4
=> x = 4/2 = 2

Hence, We can see the solution being represented on the number line.

Example 2: Represent the region which satisfy the two inequalities  4x + y $\leq$ 8 , and 15 x + 7 y > 105 in two dimensional graph.
Solution:
Step 1: Write inequalities into equations as
4x + y = 8                ---------(1) and
15 x + 7y = 105      ----------(2)
Step 2: Find the x and y intercepts of each line.

4x + y = 8
 x y 0 8 2 0

15 x + 7y = 105
 x y 0 15 7 0

Step 3: Analyze the solution area.
Since the inequality is 4x + y $\leq$ 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

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

Step 4: Design the graph.

The graphs of these two inequalities are shown below.

The shaded region represents the desired solution set.

Example 3: Solve 2x + 5 $\geq$ 13. Represent the solution on the number line, if x belongs to set of Real numbers.

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

In the above number line, the darkened portion x $\geq$ 4, is the solution set for the inequality.

Practice Questions

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 $\leq$ 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 $\geq$ - 2.