General line equations

In all examples, write the GENERAL EQUATION OF a line that is given in some way.

A) the line is given parametrically: x = - 4 + 2p, y = 2 - 3p

B) the line is given by the slope form: y = 3x - 1

C) the line is given by two points: A [3; -3], B [-5; 2]

D) the line intersects the y-axis at point 0; 6 and has a slope k = 2

Result

p₁ = (Correct answer is: )
p₂ = (Correct answer is: )
p₃ = (Correct answer is: )
p₄ = (Correct answer is: )

Solution:

$$\begin{gathered} x=-4+2p \\ y=2-3p \\ 3x=-12+6p \\ 2y=4-6p \\ {p}_1:3x+2y+8=0 \end{gathered}$$

$$\begin{gathered} y=3x-1 \\ {p}_2:3x-y-1=0 \end{gathered}$$

$$\begin{gathered} a_x+b_y+c=0 \\ c=19 \\ a\cdot 3+b\cdot (-5)+c=0 \\ a\cdot (-5)+b\cdot 2+c=0 \\ a\cdot 3+b\cdot (-5)+19=0 \\ a\cdot (-5)+b\cdot 2+19=0 \\ 3a-5b=-19 \\ 5a-2b=19 \\ a=7 \\ b=8 \\ {p}_3:7x+8y+19=0 \end{gathered}$$

$$\begin{gathered} y=k_x+q \\ k=2 \\ 6=k\cdot 0+q \\ 6=2\cdot 0+q \\ q=6 \\ {p}_4:y=2\cdot x+6 \\ {p}_4:2x-y+6=0 \end{gathered}$$

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