# Framing a Linear Equation

## Linear equation in one variable

An equation of the form *ax + by + c* = 0, where *a, b* and *c* are real numbers, such that *a* and *b* are not both zero, is called a **linear equation in two variables**.

Linear equations in one variable, of the type *ax* + *b* = 0, can also expressed as a linear equations in two variables. Since, *ax* + *b* = 0 ⇒ *ax* + 0.*y* + *b* = 0.

A **solution** of a linear equation in two variables is a pair of values, one for *x* and one for *y*, which satisfy the equation.

- The solution of a linear equation is not effected when
- The same number is added or subtracted from both the sides of an equation.
- Multiplying or dividing both the sides of the equation by the same non-zero number

When an equation has only one variable of degree one, then that equation is known as linear equation in one variable.

- Standard form: ax+b=0, where a and b ϵ R & a ≠ 0
- Examples of linear equation in one variable are :
- 3x-9 = 0

- 2t = 5

## Linear equation in 2 variables

When an equation has two variables both of degree one, then that equation is known as linear equation in two variables.

**Standard form:** ax+by+c=0, where a,b,c ϵ R & a,b ≠ 0 Examples of linear equations in two variables are:

- 7x+y=8
- 6p-4q+12=0

# Examples of a Linear Equations

## Solution of linear equation in 2 variables

A linear equation in two variables has **infinitely many solutions**.

Every point on the line satisfies the equation of the line and every solution of the equation is a point on the line.

A linear equation in two variables is represented geometrically by a straight line whose points make up the collection of solutions of the equation. This is called the **graph** of the linear equation.

*x *= 0 is the equation of the *y*-axis and* y* = 0 is the equation of the *x*-axis.

The graph of ** x = k** is a straight line parallel to the

*y*-axis.

**For example, **the graph of the equation *x* = 5 is as follows:

A linear equation in two variables has a pair of numbers that can satisfy the equations. This pair of numbers is called as the solution of the linear equation in two variables.

- The solution can be found by assuming the value of one of the variable and then proceed to find the other solution.
- There are infinitely many solutions for a single linear equation in two variables.

The graph of ** y = k** is a straight line parallel to the

*x*-axis.

**For example,** the graph of the equation *y* = 5 is as follows:

An equation of the type ** y = mx** represents a line passing through the origin, where

*m*is a real number. For example, the graph of the equation

*y*= 2

*x*is as follows:

# Graph of a Linear Equation

## Graphical representation of a linear equation in 2 variables

- Any linear equation in the standard form ax+by+c=0 has a pair of solutions (x,y), that can be represented in the coordinate plane.
- When an equation is represented graphically, it is a straight line that may or may not cut the coordinate axes.

## Solutions of Linear equation in 2 variables on a graph

- A linear equation ax+by+c=0 is represented graphically as a straight line.
- Every point on the line is a solution for the linear equation.
- Every solution of the linear equation is a point on the line.
**Lines passing through origin** - Certain linear equations exist such that their solution is (0,0). Such equations when represented graphically pass through the origin.
- The coordinate axes x-axis and y-axis can be represented as y=0 and x=0 respectively.
**Lines parallel to coordinate axes** - Linear equations of the form y=a, when represented graphically are lines parallel to the x-axis and a is the y-coordinate of the points in that line.
- Linear equations of the form x=a, when represented graphically are lines parallel to the y-axis and a is the x-coordinate of the points in that line.