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 slope form gives the line: 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

Correct answer:

p₁ : p1:3x+2y+8=0
p₂ = 0
p₃ : p3: 7x+8y+19=0
p₄ : p4: 2x-y+6=0

Step-by-step explanation:

$$\begin{gathered} x=-4+2p \\ y=2-3p \\ 3x=-12+6p \\ 2y=4-6p \\ {p}_1:p_1:3x+2y+8=0=0:1 \end{gathered}$$

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

$$\begin{gathered} a_x+b_y+c=0 \\ c=19 \\ a\cdot 3+b\cdot (-5)+c=0 \\ a\cdot (-5)+b\cdot 2+c=0 \\ a\cdot 3+b\cdot (-5)+19=0 \\ a\cdot (-5)+b\cdot 2+19=0 \\ 3a-5b=-19 \\ 5a-2b=19 \\ Pivot:Row1\leftrightarrow Row2 \\ 5a-2b=19 \\ 3a-5b=-19 \\ Row2-\frac{3}{5}\cdot Row1\to Row2 \\ 5a-2b=19 \\ -3.8b=-30.4 \\ b=\frac{-30.4}{-3.8}=8 \\ a=\frac{19+2b}{5}=\frac{19+2\cdot 8}{5}=7 \\ a=7 \\ b=8 \\ {p}_3:p_3:7x+8y+19=0=0:1 \end{gathered}$$

$$\begin{gathered} y=k_x+q \\ k=2 \\ 6=k\cdot 0+q \\ 6=2\cdot 0+q \\ q=6 \\ q=\frac{6}{1}=6 \\ q=6 \\ {p}_4:y=2\cdot x+6 \\ {p}_4:p_4:2x-y+6=0=0:1 \end{gathered}$$

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