Question

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 1

see below for explanation

3x +2y = 15

Explanation:

The x-coordinate and the y-coordinate are both linear functions of t, so their relationship to each other is linear. A graph of those (x, y) points must be a line.

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You can find an equation by solving one of the expressions for t, then substituting that into the other.

x = 2t +3

t = (x -3)/2

Substituting into the y expression, we get ...

y = 3 -3(x -3)/2

y = (-3/2)x + 15/2

3x +2y = 15 . . . . . . . add 3/2x and multiply by 2 to put into standard form

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The attached graph shows a portion of the line defined parametrically (0 ≤ t ≤ 1). It is dotted so you can see it overlays the line defined by the linear equation in x and y.