Question 966881: Suppose you graphed every single point of the form (2t + 3, 3-3t). For example, when t=2, we have 2t + 3 = 7 and 3-3t = -3, so (7,-3) is on the graph. Explain why the graph is a line, and find an equation whose graph is this line.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you can calculate the slope using the variable of T.
let's see how you would do that>
your (x,y) point is (2T+3, 3-3T).
let (x1,y1) be equal to (2T1+3, 3-3T1).
let (x2,y2) be equal to (2T2+3, 3-3T2).
slope is equal to (y2 - y1) / (x2 - x1).
this would be equal to ( (3-3T2) - (3-3T1) ) divided by ( (2T2+3) - (2T1+3) )
simplify this equation to get:
slope = (3 - 3T2 - 3 + 3T1) divided by (2T2 + 3 - 2T1 - 3)
the 3 and the -3 in both expressions cancels out and you are left with:
slope = (-3T2 + 3T1) divided by (2T2 - 2T1)
you can factor out the common -3 in the numerator and the common 2 in the denominator to get:
slope = (-3 * (T2 - T1)) divided by (2 * (T2 - T1)).
the (T2 - T1) in the numerator and denominator cancel out and you are left with:
slope = -3/2.
that's a common slope which indicates that the graph of the equation of the points is a straight line.
to solve for the equation of the line, take 2 points and then use the point slope form of the equation to find the slope intercept form of the equation.\
the point slope form of the equation is y - y1 = m * (x - x1)
you already know the slope because you just solved for it.
the slope if -3/2.
the point slope form of the equation becomes y - y1 = -3/2 * (x - x1).
you only need one point to finalize the pokint slope form of the equation.
you can choose any point.
let t = 0, and your point would be (3,3).
that's ecause 2*0 + 3 = 3 for the x value, and 3 - 3*0 = 3 for the y value.
you make (x1,y1) equal to (3,3) and put it in the slope intercept form of the equation.
you get:
y - 3 = -3/2 * (x - 3)
you now want to convert this to slope intercept form of the equation.
add 3 to both sides of the equation to get y = -3/2 * (x - 3) + 3
simplify the right side of the equation to get y = -3/2 * x + 9/2 + 3.
simplify further to get y = -3/2 * x + 15/2.
that's your equation.
the graph of that eqution is shown below:
the point (3,3) is when T = 0.
the point (0,7.5) is when T = -3/2.
the point (5,0) is when T = 1.
if the algebra for finding the slope was too intense for you, you could have found it easiwer by simply plotting several points using various random values of T and then calculating the slope several time.
you would have found that the slope was always -3/2 which indicates that you have a straight line.
the rest would then be the same to find the equation for the straight line.
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