Monday, August 18, 2014

Coordinate Geometry


Coordinate Geometry: y=mx+b

Why Equals Am-ex Plus Bee?

Why not? This format for an x-y equation helps when solving coordinate geometry problems. It standardizes the look so it's easier to compare different equations. The equation itself often gives you hints to where the line the equation represents will sit on the graph.

Y and X are graph points, So what are M and B?

In the equation y=mx+b, y and x are the coordinates (as in the two ordered pairs/graph points shown here). m represents the slope of the line and b represents the y-intercept (the value of y when x is 0 (zero)).

In order to solve an equation in this format, you need a few pieces of information.
  • If the problem gives you m (the slope) and b (the y intercept or the value of y when x is zero), you can figure out the line with a few xs and matching ys (you chose values for x, plug it into the equation and figure out what the y value is)
  • If you are given an x and a matching y (an ordered pair) with either the slope or the y intercept, you can find the other
  • If you are given two ordered pairs (as in the graphic here), you can figure out m and b

Figuring out the Slope and Y-Intercept from Two Points

In order to solve a y=mx+b problem when you have two points, you first have to figure out the slope. The slope is defined as the "rise over run". This can also be figured as the change in y over the change of x.

As you can see in the graphic here, you decide which point is (x1, y1) and which is (x2, y2). It really doesn't matter which one you subtract from which, just designate the two points. Then, subtract y2 from y1 and divide it by x1-x2 (see the graphic). In this case, we have 0-6=-6 and 5-0=5. The slope, in this case, is -6/5. (Note: a positive slope moves from the lower left to the upper right and the negative slope moves from the lower right to the upper left.)

Figuring the Equation with One Point and the Slope

If you have the slope (m) and one point, you substitute the x and y in the equation and put the slope in where the m should be. So, in this case, we use 0 for x, 6 for y and -6/5 for m. The progression in the graphic shows how you would solve this algebraically.

Note: in this case, solving for b is even easier, since b is the value of y when x is 0 -- you can see from the x,y pair that, when x is zero, y is six.

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