Explanation:
first of all, a function is an equation.
it associates a function result variable (typically called y) with a calculation based on an input variable (usually caked x), or in more complex mathematics based on a group of input variables.
but not every equation is a function. to be a function every valid x value must have exactly one associated y value.
e.g. x = 5 is an equation. but it does not restrict y in any way. so, any value of y (infinitely many) is valid for x = 5.
therefore, this is not a function.
a linear equation/function is a straight line (hence the name).
it is characterized by an inclination (typically called slope or rate of change) and its absolute position typically incentives by the interception points particularly with the y-axis on a coordinate grid.
there are various ways to describe a line in a formal way.
the slope-intercept form :
y = ax + b
a being the slope, b being the y-intercept (y-value when x = 0).
the point-slope form
y - y1 = a(x - x1)
again, a is the slope, (x1, y1) is an identified point (coordinates) on the line.
the general (often called standard) form
gx + hy = c
g, h, c are not describing anything directly, but after transforming this standard form they build the "a" and "b" terms of the other forms.
the slope is the ratio of (y coordinate change / x coordinate change) when going from one point on the line to another. for a line this is constant for any pair of points you can pick on the line.
in other words it tells us how many units y changes, when x changes by a certain amount of units
basically, a line is the collection of all the points for which the given equation is true (when using the x coordinate in the equation we get the corresponding y as calculation result, or when using x and y of any point on the line in the equation, then the equation is true).
graph :
the slope is found by checking 2 points and calculating the y diff / x diff ratio. e.g. starting with (0, 0) if that point is in the line, we increase x by 1 and check the y value there : (1, y).
so, the slope is for that example (y - 0)/(1 - 0) = y.
the y-intercept is found by checking the y-value for x = 0.
the y-interception point is therefore (0, b).
the x-intercept is the x-value when y = 0.
the x-interception point is therefore (x-intercept, 0).
table :
for the slope we pick again 2 data points of the table and calculate y diff / x diff.
as explained this has to be constant for any picked pair of data points of the table. then you know it is a linear equation.
if we are lucky, the table contains data points with x = 0 and/or y = 0. then we have the corresponding intercept values.
but if not, we need to use the coordinates of 1 point in the e.g. slope-intercept form to create an equation with 1 variable : b. and then we solve it to get b.
equation :
we need to bring the equation into a form that we get
y = ...
then the factor of x is the slope. and the constant term (even if it is not there, as it means it is 0) defines the y-intercept
everything else is as described above
description :
the description needs to give us either data points or an indication about the slope and the y-intercept.
when we have 2 points, we can define a line through them. in other words, any pair of points defines a line.