Question 253894
a) You are correct in saying that each value is increasing by 8. Since this increase is constant (ie doesn't change), this means that we have a linear equation. So these points form a straight line. Recall that we only need two points to find the equation of the line. So let's use the first two points (0,-15) and (1,-7). You can use any two points.



First let's find the slope of the line through the points *[Tex \LARGE \left(0,-15\right)] and *[Tex \LARGE \left(1,-7\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(0,-15\right)]. So this means that {{{x[1]=0}}} and {{{y[1]=-15}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(1,-7\right)].  So this means that {{{x[2]=1}}} and {{{y[2]=-7}}}.



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



{{{m=(-7--15)/(1-0)}}} Plug in {{{y[2]=-7}}}, {{{y[1]=-15}}}, {{{x[2]=1}}}, and {{{x[1]=0}}}



{{{m=(8)/(1-0)}}} Subtract {{{-15}}} from {{{-7}}} to get {{{8}}}



{{{m=(8)/(1)}}} Subtract {{{0}}} from {{{1}}} to get {{{1}}}



{{{m=8}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(0,-15\right)] and *[Tex \LARGE \left(1,-7\right)] is {{{m=8}}}



Now let's use the point slope formula:



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



{{{y--15=8(x-0)}}} Plug in {{{m=8}}}, {{{x[1]=0}}}, and {{{y[1]=-15}}}



{{{y+15=8(x-0)}}} Rewrite {{{y--15}}} as {{{y+15}}}



{{{y+15=8x+8(0)}}} Distribute



{{{y+15=8x+0}}} Multiply



{{{y=8x+0-15}}} Subtract 15 from both sides. 



{{{y=8x-15}}} Combine like terms. 




So the equation that goes through the points *[Tex \LARGE \left(0,-15\right)] and *[Tex \LARGE \left(1,-7\right)] is {{{y=8x-15}}}



 Notice how the graph of {{{y=8x-15}}} goes through the points *[Tex \LARGE \left(0,-15\right)] and *[Tex \LARGE \left(1,-7\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,8x-15),
 circle(0,-15,0.08),
 circle(0,-15,0.10),
 circle(0,-15,0.12),
 circle(1,-7,0.08),
 circle(1,-7,0.10),
 circle(1,-7,0.12)
 )}}} 
 
 
 Graph of {{{y=8x-15}}} through the points *[Tex \LARGE \left(0,-15\right)] and *[Tex \LARGE \left(1,-7\right)]

 

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b)


Now to find the y values at x=6, 7, and 8, just plug them in to get: 


x=6: {{{y=8(6)-15=48-15=33}}}


x=7: {{{y=8(7)-15=56-15=41}}}


x=8: {{{y=8(8)-15=64-15=49}}}



So the corresponding y values for x=6, 7, and 8 are y=33, 41, and 49 respectively. Take note that 33 is 8 more than 25 (the y value for x=5), 41 is 8 more than 33, and 49 is 8 more than 41. So this supports our answer.