I have a question of how do I determine the variables when graphing linear equation? For an example graph x+2y=4, how do you come up with the variables as (0,2) (2,1) and (4,0)? The example in my book does not tell me how they came up with this. I know how to graph, I need to know how to come up with the variables.
basically, you pick any variable you want, and let the equation calculate the corresponding second variable.
e.g. pick y=2, (one of their answer points)
then x+2(2)=4
x+4=4
so x=0, then when x=0,y=2, which is the point (0,2)
(the x is always written first, the y second)
in general one would isolate one of the variables, in this case I would say
x=4-2y
Now by choosing any y value you want, you would generate a matching x value.
e.g. let y =8, then
x=4-2(8)
x=-12 --------> point would be (-12,8) etc
To determine the variables when graphing a linear equation, you need to understand that any linear equation can be written in the form y = mx + b, where m represents the slope and b represents the y-intercept.
In the example you mentioned, x + 2y = 4, you can rewrite it in the slope-intercept form by isolating y:
2y = -x + 4
y = (-1/2)x + 2
Now, you can identify the slope and y-intercept. The coefficient of x (-1/2) represents the slope (m), and the constant term (2) represents the y-intercept (b).
To graph this equation, you can start by plotting the y-intercept, which is the point where the line intersects the y-axis. In this case, the y-intercept is (0, 2). So, put a point on the y-axis at y = 2.
Next, you can use the slope to determine additional points to plot. The slope tells you how the line changes as you move horizontally and vertically. For every unit increase in x, the line goes down by 1/2 (since the slope is -1/2).
To find the next point, you can start at the y-intercept and move horizontally by 2 units (since you're increasing x by 2), and then move vertically down by 1 unit (since the slope is -1/2). This gives you the point (2, 1).
Similarly, you can find another point by starting at the y-intercept, moving horizontally by 4 units (increasing x by 4), and then moving vertically down by 2 units (according to the slope). This gives you the point (4, 0).
By plotting these three points (y-intercept, x+2y=4), you can draw a straight line that passes through all of them. This line represents all the possible solutions (x, y) that satisfy the given linear equation.
To summarize, determining the variables (x, y) when graphing a linear equation involves rewriting the equation in the slope-intercept form (y = mx + b) and then identifying the slope (m) and the y-intercept (b). Using the y-intercept and the slope, you can plot points and draw a line that represents the equation.