The steepness of a hill is called a slope. The same goes for the steepness of a line. The slope is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run.
$$slope=\frac{rise}{run}=\frac{change\: in \: y}{change \: in\: x}$$
The slope of a line is usually represented by the letter m. (x1, y1) represents the first point whereas (x2, y2) represents the second point.
$$m=\frac{y_{2}\, -y_{1}}{x_{2}\, -x_{1}}$$
It is important to keep the x-and y-coordinates in the same order in both the numerator and the denominator otherwise you will get the wrong slope.
Example
Find the slope of the line
(x1, y1) = (-3, -2) and (x2, y2) = (2, 2)
$$m=\frac{y_{2}\, -y_{1}}{x_{2}\, -x_{1}}=\frac{2-\left ( -2 \right )}{2-\left ( -3 \right )}=\frac{2+2}{2+3}=\frac{4}{5}$$
A line with a positive slope (m > 0), as the line above, rises from left to right whereas a line with a negative slope (m < 0) falls from left to right.
$$m=\frac{y_{2}\, -y_{1}}{x_{2}\, -x_{1}}=\frac{\left (-1 \right )-3}{2-\left ( -2 \right )}=\frac{-1-3}{2+2}=\frac{-4}{4}=-1$$
If two lines have the same slope the lines are said to be parallel.
You can express a linear function using the slope intercept form.
$$y=mx+b$$
$$m=slope$$
$$b=y - intercept$$
Video lesson
Find the slope
Video transcript
Does the following table represent a linear equation? So let's see what's going on here. When x is negative 7, y is 4. Then when x is negative 3, y is 3. So let's see what happened to what our change in x was. So our change in x-- and I could even write it over here, our change in x. So going from negative 7 to negative 3, we had an increase in 4 in x. And what was our change in y? And this triangle, that's just the Greek letter delta. It's shorthand for "change in." Well, our change in y when x increased by 4, our y-value went from 4 to 3. So our change in y is negative 1. Now, in order for this to be a linear equation, the ratio between our change in y and our change in x has to be constant. So our change in y over change in x for any two points in this equation or any two points in the table has to be the same constant. When x changed by 4, y changed by negative 1. Or when y changed by negative 1, x changed by 4. So we have to have a constant change in y with respect to x of negative 1/4. Let's see if this is true. So the next two points, when I go from negative 3 to 1, once again I'm increasing x by 4. And once again, I'm decreasing y by negative 1. So we have that same ratio. Now, let's look at this last point. When we go from 1 to 7 in the x-direction, we are increasing by 6. And when we go from 2 to 1, we are still decreasing by 1. So now this ratio, going from this third point to this fourth point, is negative 1/6. So it is not. So just for this last point right over here, for this last point, our change in y over change in x, or I should say, really, between these last two points right over here, our change in y over change in x-- let me clear this up. Let me make it clear. So just between these last-- in magenta. Just between these last two points over here, our change in y is negative 1, and our change in x is 6. So we have a different rate of change of y with respect to x. Because we had a different rate of change of y with respect to x, or ratio between our change in y and change in x, this is not a linear equation. No, not a linear equation.
A linear equation is an equation for a straight line
These are all linear equations:
Let us look more closely at one example:
Example: y = 2x + 1 is a linear equation:
The graph of y = 2x+1 is a straight line
- When x increases, y increases twice as fast, so we need 2x
- When x is 0, y is already 1. So +1 is also needed
- And so: y = 2x + 1
Here are some example values:
| -1 | y = 2 × (-1) + 1 = -1 |
| 0 | y = 2 × 0 + 1 = 1 |
| 1 | y = 2 × 1 + 1 = 3 |
| 2 | y = 2 × 2 + 1 = 5 |
Check for yourself that those points are part of the line above!
Different Forms
There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").
But the variables (like "x" or "y") in Linear Equations do NOT have:
- Exponents (like the 2 in x2)
- Square roots, cube roots, etc
Examples: These are NOT linear equations:
Slope-Intercept Form
The most common form is the slope-intercept equation of a straight line:
Example: y = 2x + 1
- Slope: m = 2
- Intercept: b = 1
Point-Slope Form
Another common one is the Point-Slope Form of the equation of a straight line:
y − y1 = m(x − x1) |
Example: y − 3 = (¼)(x − 2)
It is in the form y − y1 = m(x − x1) where:
- y1 = 3
- m = ¼
- x1 = 2
General Form
And there is also the General Form of the equation of a straight line:
Ax + By + C = 0 |
| (A and B cannot both be 0) |
Example: 3x + 2y − 4 = 0
It is in the form Ax + By + C = 0 where:
- A = 3
- B = 2
- C = −4
There are other, less common forms as well.
As a Function
Sometimes a linear equation is written as a function, with f(x) instead of y:
| y = 2x − 3 |
| f(x) = 2x − 3 |
| These are the same! |
And functions are not always written using f(x):
| y = 2x − 3 |
| w(u) = 2u − 3 |
| h(z) = 2z − 3 |
| These are also the same! |
The Identity Function
There is a special linear function called the "Identity Function":
f(x) = x
And here is its graph:
It makes a 45° (its slope is 1)
It is called "Identity" because what comes out is identical to what goes in:
| 0 | 0 |
| 5 | 5 |
| −2 | −2 |
| ...etc | ...etc |
Constant Functions
Another special type of linear function is the Constant Function ... it is a horizontal line:
f(x) = C
No matter what value of "x", f(x) is always equal to some constant value.
Using Linear Equations
You may like to read some of the things you can do with lines:
- Finding the Midpoint of a Line Segment
- Finding Parallel and Perpendicular Lines
- Finding the Equation of a Line from 2 Points