Method Of Substitution with 3 variables.

Nov 2015
1
0
Canada
Sorry for lack of intelligence if it doesn't meet the standards of this website, but I am only in grade 9 academic math, but I am pushing my territory further every day, while I was looking for puzzles online, I came across a puzzle for a 3D graph. This puzzle requires method of substituion, which I was thinking to just break it down into 2 variables, by subbing 1 in, and then solving those 2 seperately. Have I gone the right direction with this? Anyway how would I solve the following question?

x+y-z = 4
3x-2y+z = 15
 

romsek

MHF Helper
Nov 2013
6,725
3,030
California
since there are only two equations and 3 variables the solution set for this is actually a line of points.

$x+y-z=4$

$z = x+y-4$

$3x - 2y + z = 15$

$3x - 2y + (x+y-4) = 15$

$4x -y = 19$

$y = 4x - 19$

$z = x+y-4 = x +4x-19 - 4 = 5x - 23$

so the solution is

$x \begin{pmatrix}1 \\ 4 \\ 5\end{pmatrix} + \begin{pmatrix}0 \\ -19 \\ -23\end{pmatrix},~~x \in \mathbb{R}$
 
Feb 2014
1,748
651
United States
Sorry for lack of intelligence if it doesn't meet the standards of this website, but I am only in grade 9 academic math, but I am pushing my territory further every day, while I was looking for puzzles online, I came across a puzzle for a 3D graph. This puzzle requires method of substituion, which I was thinking to just break it down into 2 variables, by subbing 1 in, and then solving those 2 seperately. Have I gone the right direction with this? Anyway how would I solve the following question?

x+y-z = 4
3x-2y+z = 15
This says the same thing as romsek's answer, but does not use the language of linear algebra, which you may not have learned yet. You do not say what the puzzle is so we cannot help you directly with the puzzle.

Generally, if you have a system with n unknowns, you need n pieces of consistent and independent information to find the unknowns. (Consistency means that no piece of information contradicts another. Independent means that no piece of the information given can be deduced from the other pieces of information.) The most common way in which you get those n pieces of information is through n equations. However, in lots of puzzles, you are not given the information to set up n independent and consistent equations. Instead, you are given the information to set up n - 1 equations, and it is implied that the answers must be non-negative integers or the smallest possible non-negative integers. So your nth piece of information is that the unknowns are non-negative integers (or relatively small non-negative integers). We do not know what your puzzle is so that may not be relevant.

$x + y - z = 4 \implies z = x + y - 4.$

$3x - 2y + z = 15 \implies 3x - 2y + x + y - 4 = 15 \implies$

$4x - 4 - 15 = y \implies y = 4x - 19.$

Now, as romsek said, y = 4x - 19 does not describe a single point, but a line of points. So there are an infinite number of potential answers

If, however, your puzzle implies (as many puzzles do) that the desired answer involves the smallest possible non-negative integers, we can see that

$x \le \dfrac{19}{4} \implies y \le 0 \implies x \ge 5.$

$x = 5 \implies y = 1 \implies 5 + 1 - z = 4 \implies z = 2.$
 
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