# These three points are on the same line: (1,2,3),(2,4,−6),(−3,−6,x). Find x. Explain your answer fully.

Question
Vectors
These three points are on the same line: (1,2,3),(2,4,−6),(−3,−6,x). Find x. Explain your answer fully.

2021-03-12
You could do this using the vector equation of a line.
$$\displaystyle{r}{\left({t}\right)}={r}{0}+{t}{v}$$
We can use (1,2,3)(1,2,3) as the point r0 and (2,4,−6)−(1,2,3)=(1,2,−9) as the direction vector v. Then if (−3,−6,x)is on the line, it should be a solution r(t) to this equation. So let's see which xx this is possible for.
$$\displaystyle{\left(−{3},−{6},{x}\right)}={\left({1},{2},{3}\right)}+{t}{\left({1},{2},−{9}\right)}$$
$$\displaystyle{\left\lbrace-{3}={1}+{t},-{6}={2}+{2}{t},{x}={3}-{9}{t}\right\rbrace}$$
From the first equation, we see that t=−4t=−4. Hence, from the last equation, we get x=39

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1
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$$\begin{array}{|c|c|c|}\hline &\text{Less Than High School}&\text{High School}&\text{More Than High School}\\\hline \text{Better off} &140&440&430\\ \hline \text{Same as}&60&230&110\\ \hline \text{Worse off}&180&280&80\\ \hline\end{array}\\$$
$$P(\text{more than high school or worse off})=?$$