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# Slope formula

Learn how to write the slope formula from scratch and how to apply it to find the slope of a line from two points.
It's kind of annoying to have to draw a graph every time we want to find the slope of a line, isn't it?
We can avoid this by writing a general formula for slope. Before we start, let's remember how slope is defined:
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction
Let's draw a line through two general points left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis.
An expression for start color #1fab54, start text, c, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54 is start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54:
Similarly, an expression for start color #e07d10, start text, c, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10 is start color #e07d10, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10:
Now we can write a general formula for slope:
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, divided by, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, end fraction
That's it! We did it!

## Using the slope formula

Let's use the slope formula to find the slope of the line that goes through the points left parenthesis, 2, comma, 1, right parenthesis and left parenthesis, 4, comma, 7, right parenthesis.
Step 1: Identify the values of x, start subscript, 1, end subscript, x, start subscript, 2, end subscript, y, start subscript, 1, end subscript, and y, start subscript, 2, end subscript.
x, start subscript, 1, end subscript, equals, 2
y, start subscript, 1, end subscript, equals, 1
x, start subscript, 2, end subscript, equals, 4
y, start subscript, 2, end subscript, equals, 7, space, space, space, space, space, space, space, space
Step 2: Plug in these values to the slope formula to find the slope.
start text, S, l, o, p, e, end text, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction, equals, start fraction, 7, minus, 1, divided by, 4, minus, 2, end fraction, equals, start fraction, 6, divided by, 2, end fraction, equals, 3
Step 3: Gut check. Make sure this slope makes sense by thinking about the points on the coordinate plane.
Yup! This slope seems to make sense since the slope is positive, and the line is increasing.

## Using the slope formula walkthrough

Let's use the slope formula to find the slope of the line that goes through the points left parenthesis, 6, comma, minus, 3, right parenthesis and left parenthesis, 1, comma, 7, right parenthesis.
Step 1: Identify the values of x, start subscript, 1, end subscript, x, start subscript, 2, end subscript, y, start subscript, 1, end subscript, and y, start subscript, 2, end subscript.
x, start subscript, 1, end subscript, equals
y, start subscript, 1, end subscript, equals
x, start subscript, 2, end subscript, equals
y, start subscript, 2, end subscript, equals

Step 2: Plug in these values to the slope formula to find the slope.
start text, S, l, o, p, e, end text, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction, equals

Step 3: Gut check. Make sure this slope makes sense by thinking about the points on the coordinate plane.
Does this slope make sense?

## Let's practice!

1) Use the slope formula to find the slope of the line that goes through the points left parenthesis, 2, comma, 5, right parenthesis and left parenthesis, 6, comma, 8, right parenthesis.

2) Use the slope formula to find the slope of the line that goes through the points left parenthesis, 2, comma, minus, 3, right parenthesis and left parenthesis, minus, 4, comma, 3, right parenthesis.

3) Use the slope formula to find the slope of the line that goes through the points left parenthesis, minus, 5, comma, minus, 7, right parenthesis and left parenthesis, minus, 2, comma, minus, 1, right parenthesis.