Linear Equation

Linear Equations

An Overview: Linear Equations 

A Linear Equation is defined as an equation that has no other mathematical operation in it other than Addition, Subtraction and Multiplication of a variable or an element. It is not necessary that the given element or variables may not be multiplied by each other. A linear equation also does not consist of a variable less than 1.

The graph that represents a Linear Equation is always a straight line. In simple words, a linear equation is an equation or equality between two given algebra-based equations that involve some known or known values is called a linear equation. A linear equation consists of two sides, a right-hand side and a left-hand side. The basic concept that is to be followed while solving these sums is that the Right Hand Side (RHS) must be equal to the Left Hand Side (LHS).

Examples of a Linear Equation- 

10x – 3y = 7

x = 4

16p – 8q = 32

m = 12x

10x + 30y = 70

x + y = 14

6p – 18q = 2

x = 417y

Standard Form of a Linear Function 

A linear equation has a general form which is written as-

Ax + By = C, the conditions which are to be kept in mind here are that A is equal or greater than zero, A and B cannot be zero.

Therefore, 

y = -10x + 4 in the standard form will be written as 10x + y = 4

14p + 14q = 10 in the standard form will be written as 4p + 4q = 40.

And when we simplify this equation, it will become p + q = 10.

Linear Equations in Two Variables

An equation that consists of two variables and the exponents of each variable is one and has no term that involves the product of the variables is called a linear equation in two variables. 

For example, 2x + 3y = 4 and x – 2y + 2 = 3x + y + 6 are linear equations in two variables.

Forms of a Linear Equation

There are three main forms of a Linear Equation, namely

Slope-Intercept Form 

Point-Slope Form

Standard Form

Slope-Intercept Form 

The Slope-Intercept Form of the equation is used to find the slope and equation of a given linear equation when two points the line passes through are known. 

In the above-given equation, ‘m’ is the slope of the equation which we have to find and ‘b’ is the Y-Intercept.

The known common formula for a slope-intercept form is-

y = mx + b

Point-Slope Form

The Point-Slope Form of the equation is used to find the slope of a given linear equation.

In the above-given equation, ‘m’ is the slope of the equation and (x1, y1) are the known points on the line

The known common formula for a point-slope form is

y – y1= m (x – x1)

Standard Form of a Linear Equation

The standard form of a linear equation is Ax + By = C where A, B, C are constants.

The known common formula for a point-slope form is

Ax + By = C

Some Important Points to Summarize Linear Equations

1. An equation in the general form of linear equation Ax + By + C = 0 where a, b, c are considered to be real numbers. Keeping in mind that A and B are not both zero numbers is called a linear equation in two variables.
2. A linear equation in two variables has an infinite number of solutions
3. The graph that is represented by a linear equation in two variables is always a straight line.
4. In Linear Equations x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.
5. The graph of x = a is a straight line that is parallel to the y-axis.
6. The graph of y = a is a straight line that is parallel to the x-axis.
7. An equation of the order y = mx will demonstrate a line that passes through the origin.
8. Every point that is plotted on the graph related to linear equations in two variables is a solution to the given respected linear equation. In easy words, the solution of a linear equation is always a point on the graph related to the given linear equation.
9. A linear equation consists of two sides, a right-hand side and a left-hand side. The basic concept that is to be followed while solving these sums is that the Right Hand Side (RHS) must be equal to the Left Hand Side (LHS).

   Conclusion

   A linear equation is defined as an equation with a maximum degree of 1. This means that in a linear equation, no variable has an exponent greater than one.

   The solution of a linear equation is the value of the variable that makes the equation true. When the same number is added, subtracted, multiplied, or divided into both sides of a linear equation, the solution remains unchanged. A linear equation with one or two variables always has a straight line as its graph.

   To be able to solve and find solutions to Linear Equation-Based problems, one must know all the basic concepts of the topic.