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A simple example : $Z=3x_1 + 2x_2$

Z is a constant with no given value.

  • $x_1, x_2 >0 $
  • $ x_1 \leq 5$
  • $ x_2 \leq 8$

The lecturer said that the slope is $-3/2$ , which I understand. However, he stated that all lines possible using that equation are parallel to the line passing through $(0,3)$ and $(2,0)$ . I want to know how did he deduce that assumption ? I tried giving $x_1$ and $x_2$ a value of zero and solve, however $Z$ has no given value. How did he know that the line would pass through $(0,3)$ and $(2,0)$ ?

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Lines are parallel if they have the same slope. – Alfonso Fernandez Dec 29 '12 at 14:28
Yes exactly, but how do you plot them when Z is not given ? – NLed Dec 29 '12 at 14:29
You can't, but you can plot a parallel line if you pick $Z$ (or a point, that's equivalent). – Alfonso Fernandez Dec 29 '12 at 14:30
Thank you for the info – NLed Dec 29 '12 at 14:34
up vote 1 down vote accepted

He simply picked one of the infinitely many lines of slope $-3/2$: they’re all parallel to one another, so they’re all parallel to any one of them. He could just as well have picked the line through $(0,6)$ and $(4,0)$: every line of slope $-3/2$ is also parallel to that line.

He chose that specific line by setting $Z=6$: the equation is then $3x_1+2x_2=6$, so when $x_1=0$ we have $x_2=3$, and when $x_2=0$ we have $x_1=2$. He probably chose $6$ because it’s the product of the coefficients $2$ and $3$: in general the line $ax_1+bx_2=ab$ passes through the points $(b,0)$ and $(0,a)$, so the intercepts are exceptionally easy to locate.

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Perfect that makes sense. And thanks for the tip regarding why he chose 6. – NLed Dec 29 '12 at 14:33
@NLed: You’re welcome. – Brian M. Scott Dec 29 '12 at 14:34

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