Question 428604
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.