# Thread: Equation of a Line

1. ## Equation of a Line

Let me get this straight.. there are three equations:
slope - y intercept form:
y = mx + b
standard form:
ac + by + c = 0
point- slope form:
y - y1 = m(x + x1)

Now: how do you convert from one form to another form? And I'm not sure I quite understand what each equation represents. For example, in standard form, what does C represent? In slope-y intercept form, what does x represent? This is so confusing.
And the thing is, I know the information and the principles, but I can't apply it in actual math questions. Does anybody have any hints on how to learn when to use one equation rather than another, or how to solve problems with these equations?
Thanks

2. Originally Posted by ifailatmath
Let me get this straight.. there are three equations:
slope - y intercept form:
y = mx + b
standard form:
ac + by + c = 0
point- slope form:
y - y1 = m(x + x1)

Now: how do you convert from one form to another form? And I'm not sure I quite understand what each equation represents. For example, in standard form, what does C represent? In slope-y intercept form, what does x represent? This is so confusing.
And the thing is, I know the information and the principles, but I can't apply it in actual math questions. Does anybody have any hints on how to learn when to use one equation rather than another, or how to solve problems with these equations?
Thanks
I'm sure for the standard form you MEANT to write $\displaystyle ax + by + c = 0$

Consider the following:

$\displaystyle ax + by + c = 0$

$\displaystyle by = -ax - c$

$\displaystyle y = \frac{-a}{b}x - \frac{c}{a}$

That is the same as slope-intercept form! The gradient of the line is given by $\displaystyle m = \frac{-a}{b}$, and the y-intecept is the point $\displaystyle \big(0, \frac{-c}{a}\big)$.

The point-slope from comes from the fact that the gradient of a line, m, is defined as:

$\displaystyle m = \frac{y - y_1}{x - x_1}$

$\displaystyle y - y_1 = m(x-x_1)$

3. Originally Posted by ifailatmath
Let me get this straight.. there are three equations:
slope - y intercept form:
y = mx + b
standard form:
ac + by + c = 0 <--This should be ax + by + c = 0
point- slope form:
y - y1 = m(x + x1) <-- this should actually be
$\displaystyle (x-x_1)$

First, think about why the name of each formula is called that.
Slope-intercept gives you exactly what it says it does, a slope (m) and a y intercept (b)
Point-slope also gives you what it says it does,
$\displaystyle y-y_1$ are two y coordinates for two points somewhere on your line and $\displaystyle x-x_1$ are the two x-coordinates of those same y-coordinates of the points on your line. The slope IS already in the point-slope equation, you just haven't solved for it yet. It is a good equation to use if you are given the graph of a line and asked to find the equation of it. Look closer at the point-slope equation. If you got "m" by itself by dividing both sides by
$\displaystyle (x-x_1)$ that would be your slope right? $\displaystyle (x-x_1)$ is the "run" part of your slope and $\displaystyle (y-y_1)$ is the "rise" part. $\displaystyle \frac{change in y values}{change in x values}$

1) How do you convert from one form to another form?
I personally like slope-intercept form best. It is easy to work with because it tells you exactly where your line hits the y-intercept or "b" in that equation. You can get all of these into slope-intercept form by solving for "y"
a) ax + by + c = 0
by = -ax - c
y = \frac{-ax}{b}-\frac{c}{b}
***a, b & c are just coefficients (the numbers in front of your variables x & y
2) And I'm not sure I quite understand what each equation represents. For example, in standard form, what does C represent?
c is just a number, since it does not have an "x" or "y" attached to it, it will probably be part of your y-intercept. If the letters confuse you, just replace them with a number, any number. That is what is neat about algebra, regardless of what numbers are in your equation you can solve for an answer. While you are trying to understand the concept, try to use numbers that will reduce each other, like 2x + 8y + 10 = 0 Solve for y in that equation and it will put it into slope-intercept form for you.
3) In slope-y intercept form, what does x represent?
Look at the graph of y = x. It is just a line through (0,0) on your graph right? y = x is actually in slope-intercept form y = 1x + 0 The slope is understood to be 1, so we leave it out and since the graph intercepts the y axis at 0, we leave that off too!
This is so confusing. You'll get it! Hang in there! It takes practice!!

4) Does anybody have any hints on how to learn when to use one equation rather than another, or how to solve problems with these equations? When you look at a problem you are given, try not to be overwhelmed with all the "stuff" in it. See if you can get "y" by itself using addition, subtraction, multiplication or division. Then once you get y by itself, see if it looks like one of the line equations, slope-intercept, point- slope or standard