# Confused with solving a three variable equation

Here is the trouble:

A dietitian has 3 supplement items that can be made use of.

Each dosage of Supplement I has 5 devices of Vitamin B, 4 devices of calcium and also 4 devices of iron.

Each dosage of Supplement II has 2 devices of Vitamin B, 3 devices of calcium and also 2 devices of iron.

Each dosage of Supplement III has 1 device of vitamin B, 5 devices of calcium and also 2 devices of iron.

If the person requires 13 devices of Vitamin B, 16 devices of calcium and also 12 devices of iron weekly, the amount of dosages of each supplement should the dietitian suggest?

I've obtained: ¢

$5x+2y+Z=13$¢¢ $4x+3y+5z=16$ ¢¢ $4x+2y+2z=12$ ¢¢ Then: I have $x=1+z,\quad y=4-3z,\quad$ and also $(1+z,4-3z,z)$ ¢

Now I'm perplexed on just how to analyze this right into dosages, if this is the appropriate strategy. Please aid.

Sure, you can write it as a system of formulas: $$\begin{cases} 5x + 2y + z = 13 \\ 4x + 3y + 5z = 16 \\ 4x + 2y + 2y = 12 \end{cases}$$ where $x, y, z$ is the variety of dosages of supplement I, II, III, specifically.

Usage Gaussian removal to locate a remedy (if there is one). That, is, write it on matrix kind and also execute row procedures: $$ \left( \begin{array}{ccc|c} 5 & 2 & 1 & 13 \\ 4 & 3 & 5 & 16 \\ 4 & 2 & 2 & 12 \end{array} \right) \sim \left( \begin{array}{ccc|c} 5 & 2 & 1 & 13 \\ 0 & \frac{7}{5} & \frac{21}{5} & \frac{28}{5} \\ 0 & \frac{2}{5} & \frac{6}{5} & \frac{8}{5} \end{array} \right) $$

and more

EDIT: Missed that you currently had the remedy! You can connect in values of $z$, the wanted quantity of dosages of supplement III, in the remedy $(1+z, 4-3z, z)$. $z=0$ returns $(1, 4, 0)$ and also $z=1$ returns $(2, 1, 1)$ - the first number is the quantity of dosages of supplement I, the 2nd supplement II and also the 3rd supplement III.

If you connect in $z=2$ you get $(3, -2, 2)$, which have adverse dosages of supplement II. This will certainly take place for all $z \geq 2$, so these remedies are disregarded.

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