A linear equation is one that relates the first power of two variables, x and y, and its graph is always a straight line. The standard form of such an equation is
Ax + By + C = 0
where A, B and C are constants.
Every straight line has slope, usually designated by the letter m. Slope is defined as the change in y divided by the change in x between any two points (x1, y1) and (x2, y2) on the line.
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m = ∆y/∆x = (y2 - y1) ÷ (x2 - x1)
If the line passes through point (a, b) and any other random point (x, y), slope can be expressed as:
m = (y - b) ÷ (x - a)
This can be simplified to produce the slope-point form of the line:
y - b = m(x - a)
The y-intercept of the line is the value of y when x = 0. The point (a, b) becomes (0, b). Substituting this into the slope-point form of the equation, you get the slope-intercept form:
y = mx + b
You now have all you need to find the slope of a line with a given equation.
General Approach: Convert from Standard to Slope-Intercept Form
If you have an equation in standard form, it takes just a few simple steps to convert it to slope intercept form. Once you have that, you can read slope directly from the equation:
Write the Equation in Standard Form
Ax + By + C = 0
Rearrange to Get y by Itself
By = -Ax - C
y = -(A/B)x - (C/B)
Read Slope from the Equation
The equation y = -A/B x - C/B has the form y = mx +b, where
m = -(A/B)
Example 1: What is the slope of the line 2x + 3y + 10 = 0?
In this example, A = 2 and B = 3, so the slope is -(A/B) = -2/3.
Example 2: What is the slope of the line x = 3/7y -22?
You can convert this equation to standard form, but if you're looking for a more direct method to find slope, you can also convert directly to slope intercept form. All you have to do is isolate y on one side of the equal sign.
Add 22 to Both Sides and Put the y Term on the Right
3/7y = x + 22
Multiply Both Sides by 7
3y = 7x + 154
Divide Both Sides by 3
y = (7/3)x + 51.33
This equation has the form y = mx + b, and
m = 7/3