Calculating points on a line
I need to calculate the points on a line, but I only know...
- the y-intercept (e.g. 20)
- the sum of the first twelve y-values (e.g. 100)
In other words, I want to calculate the y-values at x=1...x=12
First of all, e.g. means "for example", implying that it is an example not necessarily applicable to your question. If you mean the y-intercept is at (0,20), then you can just state the coordinate. Also, the word "first" is a bit ambiguous, but you stated x = 1 through 12, so I will assume you mean the corresponding y-values of x-values 1 through 12 equals 100. You could say the first twelve natural numbers for slightly more clarity, but it's clear enough to answer.
Since it is a linear equation, the y-value must increase at a constant rate. So far, we know that the equation must look like y=mx + 20. At x=1, y=20+m where m is the slope. Then the coordinates must be (1, 20+m); (2, 20+2m); (3, 20+3m)...(12, 20 +12m). Using the rules of arithmetic series, (m+20) + (2m+20) + (3m+20)+...+(12m+20), we know that this equals (((m+20) + (12m+20))*12)/2 which equals (13m+40)*6.
According to the given information, this expression must equal 100, so we set it equal to 100. Dividing both sides by 6, and we get 13m +40 =(100/6). Then subtract 40 to get -(140/6). Divide both sides by 13 and simplify to get -(70/39) which equals -1.794871 with all decimal places repeating. So the slope is -70/39
Our line should be y = -(70/39)x + 20. Let's calculate the sum of the corresponding y-values of the x-values 1 through 12 to make sure. The sum of the arithmetic series should be: (710/39) + (640/39) + (570/39)+...+(-60/39). The first term is (710/39) and the last term is (-60/39) and their sum is (650/39). The expression should be multiplied by 12, and divided by 2 which simplifies to 6. (650/39)*6 = (3900/39)=100. If I interpreted your information correctly, the line should be y = -(70/39)x + 20. If you want to find specific points on the line, simply plug-in the desired x-value into the equation.
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