Question

Find the equation of the line passing through the points (2,11) and (-8,-19)

Answer 1

Answer: y=3x+5

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Answer 2

Slope-intercept form: y=3x+5

Standard form: 3x-y = -5

Explanation:

The equation of a line is dependent can be defined by one of two ways:

• The slope, and one point that it passes through, OR
• The slope, and its y-intercept (a more specific version of the previous, it's a point where x=0)

In either case, the first step it finding the slope!

• The formula for finding the slope between two lines is (y2-y1)/(x2-x1), or change in y divided by change in x.
• This this example, the change in y would be (-19-11) = -30
• The change in x would be (-8-2) = -10
• So change in y divided by change in x would be -30/-10 = 3 (the slope). We use the variable m to represent the slope, so we would say m=3.

If we use the more general approach, we need one more equation: this is known as the point-slope formula. This is the formula in which we can plug in a known point and a known slope to find the formula of the line defined by that point and slope.

We can use either point given! (2,11) or (-8,-19). It's up to you! I like to use the simpler/smaller numbers, so I'll use (2,11).

So, to recap:

• m=3
• point: (2,11), or in other words, x1=2, y1=11

Point-Slope formula: y-y1=m(x-x1) where y and x stay as variables, and y1, x1, and m are replaced by our values. This gives us:

• y-11=3(x-2)

Where you go from here depends on the equation form in which your text/curriculum/instructor expected the final answer. Here are a couple common forms:

Slope-intercept form: y=mx+b

• For this form, we would distribute the slope 3, then solve for y:
• y-11 = 3x-6 (distributed 3 across the parentheses)
• y=3x+5 (added 11 to both sides)

Standard form: ax+by=c

This one is a little more tricky. First distribute, then move x and y to the same side, and the constant (unchanging number) to the other, making sure has a positive, whole coefficient (number multiplying in front of it). There are several ways to do this. Here's one:

• y-11 = 3x-6 (distributed 3 across the parentheses)
• 3x-6-y = -11 (subtracted by from both sides, so that x is positive)
• 3x-y = -5 (added 6 to both sides)