for example: x+2y=6
How would you make those into coordinates on a graph?
for example: x+2y=6
How would you make those into coordinates on a graph?
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)