# Equation of line and its points

In the xy coordinate system if (a,b) and (a+3,b+k) are two points on the line defined by equation (the equation is kind of faded in text , but it seems to be like x=3y-7) then k =

A)9 , B)3 , C)1 , D)1 (Ans is 1)

Any suggestions on how that answer was calculated ?

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## 1 Answer

The slope of the line through $(a,b)$ and $(a+3, b+k)$ is $\frac{b+k-b}{a+3-a}$, which is $\frac{k}{3}$.

The slope of the line $x=3y-7$ is $\frac{1}{3}$. This is because the equation can be rewritten as $3y=x+7$, and then in standard slope-intercept form as $y=\frac{1}{3}x+\frac{7}{3}$.

These slopes are equal $\frac{k}{3}$ and $\frac{1}{3}$ are equal.

Another way: Because $(a,b)$ is on the line, we have $a=3b-7$. Because $(a+3,b+k)$ is on the line, we have $a+3=3(b+k)-7$, that is, $a+3=3b+3k-7$.

Since $a=3b-7$, we conclude that $3=3k$.

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so the answer was obtained by comparing the two slopes , right ? – MistyD Aug 28 '12 at 5:14
I was having difficulty computing the slope from the equation. Thanks for clearing that out – MistyD Aug 28 '12 at 5:16
@MistyD: Yes. I added another way of doing it. You are probably at this time learning about equations of lines and slopes, so probably the slope way is the way you are intended to do it. But the second way works fine too. – André Nicolas Aug 28 '12 at 5:18