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How do you make a graph out of an equation? (x,y)

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2 Answers

Linear equations are the first step to understanding how graphing works. The default equation which will shortly become permanent in your knowledge of math is: y = mx + b

This equation is the slope-intercept form for straight lines where 'm' represents the slope of the line, and 'b' represents the y-axis intercept (when x = 0, y = b).

Using your example, let's first change the equation to be in the slope-intercept form.

x + 2y = 6       [beginning equation]

2y = -x + 6      [subtracting 'x' from each side]

y = -x/2 + 6/2 [dividing each side by 2]

y = -1/2x + 3  [simplify]

Now the equation is in slope-intercept form. From the equation now, we can tell that the slope (m) is -1/2 and the y-axis intercept (b) is 3.

An easy first point is the intercept, so (0,3) could be your first point. From there you can either draw a straight line at a slope of negative 1/2 (which is to say for every 2 points you move right on the x-axis, you'll drop down 1 on the y-axis). Or you can input any value for x and find it's corresponding point on y by solving the equation. Making a table can help with that:
__x__|__y__

   0         3

   2         2

   6         0

*a few examples, but any 'x' value would be fine.

First, solve the given equation for y:

     x + 2y = 6

     2y = -x + 6 

     y = (-x + 6)/2

     y = (-x/2) + (6/2)

     y = (-1/2)x + 3

This form of a linear equation is know as the slope-intercept form.

Slope-intercept form:     y = mx + b ,  where m is the slope of the line and b is the y-intercept (the point where the line crosses the y-axis, or when x=0).

Thus, for the line of the equation   y = (-1/2)x + 3 :

     m = slope = Δy/Δx = rise/run = -1/2

     b = y-intercept = 3    ==>    y = 3   when   x = 0    ==>    (0, 3) 

After plotting the y-intercept, (0, 3), use the slope of the line, -1/2,  to plot other points.

That is, form the point (0, 3) go down the y-axis 1 point (since the rise part of the slope is negative) and 2 points to the right (since the run part of the slope is positive) which gives you the next set of points to the right of the y-intercept, that being (2, 2). Conversely, you can go up one point on the y-axis and 2 points to the left which gives you the next set of points to the left of the y-intercept, that being (4, -2).

Another way to do this is to simply pick a few x-values and plug them into the linear equation we found to generate the y-value for each x-value, which would give you a few x,y-coordinates to plot. For instance, for the given equation     y = (-1/2)x + 3

     when x = -4,   y = (-1/2)(-4) + 3 = 2 + 3 = 5   ==>     (-4, 5)

     when x = -2,   y = (-1/2)(-2) + 3 = 1 + 3 = 4   ==>     (-2, 4)

     when x = 0,    y = (-1/2)(0) + 3 = 0 + 3 = 3     ==>     (0, 3)

     when x = 2,   y = (-1/2)(2) + 3 = -1 + 3 = 2     ==>     (2, 2)

     when x = 4,   y = (-1/2)(4) + 3 = -2 + 3 = 1     ==>     (4, 1)