Linear Equations and Lines
Lines are defined by its slope and any point on that line (usually given as the y-intercept, but not always).
Slope is a measure of a line's steepness, relative to the horizontal.
Horizontal lines have slope zero. Vertical lines have undefined (or "no") slope, not to be confused with
zero slope. Lines that go "up" as you read left to right have positive slope, lines that go down have
negative slope.
In general, slope is a ratio of how much y changes over how much x changes. If given two points (a,b)
and (c,d), they make a line, and their slope is
and (7, 8) is
଼ିሺିହሻ
଻ିଷ
଼ାହ
= ଻ିଷ =
ௗି௕
௖ି௔
. For example, the slope of the line that connects (3, -5)
ଵଷ
ସ
.
Be careful with negative signs, and be consistent in which order you subtract.
A generic line has equation y = mx + b. m is the slope, b is the y-intercept (0,b). For example, y = 5x - 8
has a slope 5, and a y-intercept of -8 (or the point (0,-8)). So you go down to -8 on the y-axis, then you go
up 5 and over 1 (slope = 5 = 5/1), draw another point, and boom, you have the line!
In general, y-intercepts occur where x = 0, and x-intercepts occur where y = 0. Thus, in the line y = 5x - 8,
the y-intercept is (0,-8), and the x-intercept is when y = 0, so we get 0 = 5x - 8. Solve for x, you get
x = 8/5. Thus, the point (8/5, 0) is the x-intercept.
Horizontal lines have equations y = k, where k is the point at which the horizontal line crosses the y-axis.
It has no x-intercept (unless y = 0 already). Slope = 0.
Vertical lines have equations x = k, where k is the point on the x-axis where the vertical line passes
through. It has no y-intercept (unless it is x = 0 already). Slope is undefined.
Lines are sometimes written like 5x + 2y = 10. The x-intercept is where y = 0, so plug in y = 0, you get
5x = 10, so x = 2, and the x-intercept is (2,0). The y-intercept is (0,5). Can you see how I got that? With
ହି଴
ହ
those two points, the slope is ଴ିଶ = − ଶ.
Practice reading slopes off a graph too. Try to find two points then use the formula on it.
Inequalities
For a single variable, solve for the variable, and then look carefully whether you are shading left (<) or
right (>), and whether you include the point (< or >).
For two variables, plot the graph, and then select a test point not on the graph. Plug it into the variables. If
true, shade the side with the test point. If false, shade the other side.
Be aware of cases where the variable term drops out entirely. In such a case, you may have no solution, or
all real numbers may be the solution.
For example: 3(x + 2) > 3x + 6 .... when you simplify, everything cancels and you are left with 0 > 0.
That is false! (do you see why?). Thus, the solution set is empty. Now suppose we had 3(x + 2) > 3x + 6.
In this case, we get 0 > 0, which is true. In this case, all x (all real numbers) is your solution.
Exponentials
The general form is ‫ܽ = ݕ‬ሺܾሻ௫ , where b is the base. If you are given an initial quantity and a rate of
growth as a percentage, then the initial quantity is a, and b is 1 + r, where r is the growth rate (in decimal
form). For example, if a city had 50,000 people in 2010 and is growing at 5% per year, then the
population function is ‫ = ݕ‬50,000ሺ1.05ሻ௫ , where x is in years.