Linear equations are one degree equations having the exponent on the variable one. 
Working on linear algebra, we must have come across many algebra equations. These expressions have one, two or more variables. In this section will study about how to write standard equations for linear equations with variable one, two and three along with the help of some solved examples. After practice below listed problems, you will be ready to convert any equation into standard from at our own pace.


Standard Form of a Linear Equation in One Variable

An equation consists of real numbers and multiple of a variable. A linear equation with one variable is a first-degree equations because the exponent on the variable is always 1. Solution for such equations be always a value which satisfy the given equation. For the above equation, x = -B/A is the required result. 

This is the standard form of linear equations in one variable :

Ax + B = 0
Where A, and B are all real numbers and A can not be zero.

Standard Form of Linear Equation in Two Variables

Often when we deal with questions based on linear question, the very first question arise in our mind is, What is the linear standard form? The standard form of linear equations in two variables:

Ax + By = C

Where A, B And C are all real numbers and A and B can not be zero.

We can graph such equations in xy plane after finding the x-intercept and y-intercepts of the corresponding equation. For example, if we are given with the equation 2x - y = 10 then

 To Find x-intercept   Set y = 0    2x - y = 10 
2x = 10
x = 5 
  x-intercept = (5, 0)        
 To Find y-intercept  Set x = 0  2(0) - y = 10
- y =10
y = -10
 y-intercept = (0, -10)

Graph will looks like:
Standard Form of Linear Equations

Standard Form of Linear Equation in Three Variables

The standard form of linear equations in 3 variables :

Ax + By + Cz = D

Where A, B, C and D are all real numbers and A, B and C can not be zero.
Since this equation contains 3 variables, so we need 3 more similar equations to find the values of x, y and z. Mostly matrix method helps to get the result.

Examples

Example 1: What is the x-intercept and y-intercept of the line 3x - 10y = 30?
Solution: 
Given equation is 3x - 10y = 30
Find y-intercept:
Set value of x as zero
3 $\times$ 0 - 10 y = 30
- 10y = 30
Divide each side by -10
$\frac{-10y}{-10}$ = $\frac{30}{-10}$
y = -3
Find x-intercept:
Set value of y as zero
3x - 10 $\times$ 0 = 30
3x - 0 = 30
3x = 30
Dividing both sides by 3
3x/3 = 30/3
x = 10
Answer:
x-intercept is (10, 0)
y-intercept is (0, -3)

Example 2: Write equation into standard form: y = 3x/11 + 10
Solution: 
Given equation is  y = 3x/11 + 10
As above linear equation consists two variables x and y, it shows this is a linear equations with two variable. 
So convert given equation into its standard from which is ax + by + c = 0.
Now: 
Reduce the fraction
y = 3x/11 + 10
Multiply each side by 11
11(y = 3x/11 + 10)
11y = 11 $\times$ 3x/11 + 11 $\times$ 10
11y = 3x + 110
11y - 3x - 110 = 0
or -3x + 11y - 110 = 0
Which is the required result.
 -3x + 11y - 110 = 0 similar to ax + by + c = 0, where a = -3, b = 11 and c = -110

Practice Problems

Practice below listed problems and check your knowledge.
Problem 1:
Write equation into standard form: x = 4/13 y + 11
Problem 2: Find x and y intercepts of the equation 5x - 9y/3 - 16 = 0
Problem 3: Write below equations in standard from
a) x(1/2 - 3) + 2(x-4) = 0
b) 3x-  4(y + 2) - 1/2(z - x - y) + 13 = 0
c) 9y/5 + 5(x - y + 2) - 11/5 = 0