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.

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.

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.

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

When x = 0, y = 8 - 4 ( 0 ) = 8 - 0 = 8

The

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:

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.

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

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.

=> 2x + 5 - 5 = 9 - 5

=> 2x = 4

=> x = 4/2 = 2

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

4x + y = 8 ---------(1) and

15 x + 7y = 105 ----------(2)

4x + y = 8

x |
y |

0 |
8 |

2 |
0 |

15 x + 7y = 105

x |
y |

0 |
15 |

7 |
0 |

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.

The graphs of these two inequalities are shown below.

The shaded region represents the desired solution set.

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.

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.