Question 127128
First find the slope through (-4, -7) and (1, 3)


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Let's denote the first point (-4,-7) as *[Tex \Large \left(x_{1},y_{1}\right)]. In other words, *[Tex \LARGE x_{1}=-4] and *[Tex \LARGE y_{1}=-7]


Now let's denote the second point (1,3) as *[Tex \Large \left(x_{2},y_{2}\right)]. In other words, *[Tex \Large x_{2}=1] and *[Tex \Large y_{2}=3]




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{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula


{{{m=(3--7)/(1--4)}}} Plug in {{{y[2]=3}}},{{{y[1]=-7}}},{{{x[2]=1}}},{{{x[1]=-4}}}



{{{m=10/5}}} Subtract the terms in the numerator {{{3--7}}} to get {{{10}}}.  Subtract the terms in the denominator {{{1--4}}} to get {{{5}}}

  

{{{m=2}}} Reduce



So the slope of the line through the points (-4,-7) and (1,3) is {{{m=2}}}



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Now let's find the slope through (2, 6) and (4, 10)





Let's denote the first point (2,6) as *[Tex \Large \left(x_{1},y_{1}\right)]. In other words, *[Tex \LARGE x_{1}=2] and *[Tex \LARGE y_{1}=6]


Now let's denote the second point (4,10) as *[Tex \Large \left(x_{2},y_{2}\right)]. In other words, *[Tex \Large x_{2}=4] and *[Tex \Large y_{2}=10]




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{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula


{{{m=(10-6)/(4-2)}}} Plug in {{{y[2]=10}}},{{{y[1]=6}}},{{{x[2]=4}}},{{{x[1]=2}}}



{{{m=4/2}}} Subtract the terms in the numerator {{{10-6}}} to get {{{4}}}.  Subtract the terms in the denominator {{{4-2}}} to get {{{2}}}

  

{{{m=2}}} Reduce


  
So the slope of the line through the points (2,6) and (4,10) is {{{m=2}}}



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Since the slope through the two pairs of points is {{{m=2}}}, this means that the two lines are parallel