SOLUTION: What are the similarities and differences between functions and linear equations, and are all linear equations functions, and is there an instance which a linear equation is not a

Question 428604: What are the similarities and differences between functions and linear equations, and are all linear equations functions, and is there an instance which a linear equation is not a function? (Support Answer)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
a linear equation is an equation of a straight line.

as such all linear equations will be functions because there can only be 1 and only 1 value of y for each value of x.

there is, i believe, one major exception.

the equation of x = c is a linear equation because it is an equation of a straight line.

this equation is not a function because you can have more than 1 value of y for each value of x.

the equation of x = c is a vertical line.

for example:

the equation x = 5 yields one value of x and an infinite number of possible values for y.

any point on the vertical line is a potential value of y.

other than that, i believe all other straight lines other than the vertical line will be functions because, other than the vertical line, there can only be 1 value of y for each value of x.

the equation of a horizontal line is y = c, where c is a constant, such as 0, 1, 2, 3, 4, etc.

with this equation, all values of y will be the same for each value of x, but each value of x yields one and only 1 value of y.

for example, if the equation is y = 5, then when x = 1, y = 5, when x = 2, y = 5, etc.

all values of y will be 5 but each value of x yields one and only one value of y so the linear equation is a function.

any slope other than 0 will yield one and only 1 value of y.

for example, if the equation is y = x, then when x = 1, y = 1, when x = 2, y = 2, etc.

there is only 1 possible value of y for each value of x.

the only exception, as noted earlier, is when the line is vertical. then the linear equation (it still is a linear equation) is not a function, and only then.

y = |x| is still a linear equation, although it is technically not an equation of a straight line because it does change direction as x transitions from positive to negative.

it is still a function because there is still one and only one value of y for each and every value of x.

y = |x| means y = absolute value of x which means that y will always be positive.

if x is positive, then y equals the value of x.
if x is negative, then y equals the negative value of x.

this results in y always being positive.

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