A linear equation is an equation for a straight line
These are all linear equations:
yes y = 2x + 1
yes 5x = 6 + 3y
yes y/2 = 3 − x
Let us look more closely at one example:
Example: y = 2x + 1 is a linear equation:
line on a graph
The graph of y = 2x+1 is a straight line
When x increases, y increases twice as fast, so we need 2x
When x is 0, y is already 1. So +1 is also needed
And so: y = 2x + 1
Here are some example values:
xy = 2x + 1
-1y = 2 × (-1) + 1 = -1
0y = 2 × 0 + 1 = 1
1y = 2 × 1 + 1 = 3
2y = 2 × 2 + 1 = 5
Check for yourself that those points are part of the line above!
Different Forms
There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").
Examples: These are linear equations:
yes y = 3x − 6
yes y − 2 = 3(x + 1)
yes y + 2x − 2 = 0
yes 5x = 6
yes y/2 = 3
But the variables (like "x" or "y") in Linear Equations do NOT have:
Exponents (like the 2 in x2)
Square roots, cube roots, etc
Examples: These are NOT linear equations:
not y2 − 2 = 0
not 3√x − y = 6
not x3/2 = 16
Slope-Intercept Form
The most common form is the slope-intercept equation of a straight line:
y=mx+b graph
Equation of a Straight Line y=mx+b
Slope (or Gradient)Y Intercept
Example: y = 2x + 1
Slope: m = 2
Intercept: b = 1
Animation
Play With It !
You can see the effect of different values of m and b at Explore the Straight Line Graph
Point-Slope Form
Another common one is the Point-Slope Form of the equation of a straight line:
y − y1 = m(x − x1)
Point-Slope Form
Example: y − 3 = (¼)(x − 2)
It is in the form y − y1 = m(x − x1) where:
y1 = 3
m = ¼
x1 = 2
General Form
And there is also the General Form of the equation of a straight line:
Ax + By + C = 0
(A and B cannot both be 0)
Example: 3x + 2y − 4 = 0
It is in the form Ax + By + C = 0 where:
A = 3
B = 2
C = −4
There are other, less common forms as well.
As a Function
Sometimes a linear equation is written as a function, with f(x) instead of y:
y = 2x − 3
f(x) = 2x − 3
These are the same!
And functions are not always written using f(x):
y = 2x − 3
w(u) = 2u − 3
h(z) = 2z − 3
These are also the same!
The Identity Function
There is a special linear function called the "Identity Function":
f(x) = x
And here is its graph:
Identity Function
It makes a 45° (its slope is 1)
It is called "Identity" because what comes out is identical to what goes in:
InOut
00
55
−2−2
...etc...etc
Constant Functions
Another special type of linear function is the Constant Function ... it is a horizontal line:
Constant Function
f(x) = C
No matter what value of "x", f(x) is always equal to some constant value.
Using Linear Equations
You may like to read some of the things you can do with lines:
Finding the Midpoint of a Line Segment
Finding Parallel and Perpendicular Lines
Finding the Equation of a Line from 2 Points
Source: https://www.mathsisfun.com/algebra/linear-equations.html
Step-by-step explanation:
hee hmu ;) <333333333
