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Two lines are parallel if their slopes are identical. For this exercise, we have been given two points on each line, so we have enough information to calculate their slopes using the slope formula. $m=x_{2}−x_{1}y_{2}−y_{1} $
Note that when choosing points to substitute for $(x_{1},y_{1})$ and $(x_{2},y_{2}),$ it doesn't matter which points on the line you choose, since the result will be the same. Let's start with line $1,$ which passes through $(-1,1)$ and $(2,-2).$
Thus, the slope of line $1$ is equal to $1.$ We will use the same method to calculate the slopes of lines $2$ and $3.$

$m=x_{2}−x_{1}y_{2}−y_{1} $

$m=2−(-1)-2−1 $

Evaluate right-hand side

$m=1$

Line | Points | Slope |
---|---|---|

$2$ | $(2,3)$ & $(-2,7$ | $-2−27−3 =-2$ |

$3$ | $(0,-2)$ & $(5,-12)$ | $5−0-12−(-2) =-2$ |

Now that we've identified the slope of each line, we can see that line $2$ and $3$ have the same slope but not line $1.$ Therefore, the three lines are not all parallel.