How do you know if an equation is linear or non-linear?

An equation is considered linear, if it is in the form of

#y=mx+b#

where #m# is the slope of the equation, and #b# is the y-intercept.

Notice how here, #x# can only be to the power of #1#.

In here, the conditions are just simply: #m,b\inRR#

Some examples include #y=5x+4#, #y=x-2#, #y=0#, and even some like #x=1#.

Let me graph them so that you can see:

#y=5x+4#
graph{5x+4 [-10, 10, -5, 5]}

#y=x-2#
graph{x-2 [-10, 10, -5, 5]}

#y=0#

#x=1#

As you can see here, all of the following equations are represented using a straight line. An equation is considered "non-linear" is when it is not graphed using straight lines. Some examples include #y=3x^2+1#, #y=2x^3-3#, #y=x^5+43#.

See what they have in common? They all have their first #x# with a power greater than #1#.

When we graph them, they are not going to be a straight line.

#y=3x^2+1#

graph{3x^2+1 [-10, 10, -5, 5]}

#y=2x^3-3#

graph{2x^3-3 [-10, 10, -5, 5]}

#y= x^5+43#

graph{x^5+43 [-22.79, 28.18, 29.24, 54.72]}

In conclusion, a linear equation will always be in the form of #y=mx+b#, where #m# is the slope of the equation, and #b# is the y-intercept of the equation.