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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 $ℓ_{1},$ which passes through $(0,-2)$ and $(-6,1).$
The slope of line $ℓ_{1}$ is $-21 .$ We will use the same method to identify the slope of line $ℓ_{2}.$

$m=x_{2}−x_{1}y_{2}−y_{1} $

$m=-6−01−(-2) $

SubNeg$a−(-b)=a+b$

$m=-6−01+2 $

AddSubTermsAdd and subtract terms

$m=-63 $

MoveNegDenomToFracPut minus sign in front of fraction

$m=-63 $

ReduceFrac$ba =b/3a/3 $

$m=-21 $

Line | Points | $x_{2}−x_{1}y_{2}−y_{1} $ | Slope | Simplified Slope |
---|---|---|---|---|

$ℓ_{1}$ | $(0,-2)&(-6,1)$ | $-6−01−(-2) $ | $-63 $ | $-21 $ |

$ℓ_{2}$ | $(4,1)&(0,3)$ | $0−43−1 $ | $-42 $ | $-21 $ |

Now that we've identified the slope of each line, we can see that $ℓ_{1}$ and $ℓ_{2}$ have the same slope, so they are *parallel.*