Question: Find the value of a which allows solutions to the system of equations:
$4x −\; y + 2z = \;\; 7 \qquad\; [1]$
$\;\; x + \;y − 3z = −3 \qquad [2]$
$2x − 3y + 8z = \; a \qquad\; [3]$
Solution: First of all, take a look at equations $[1],[2],[3]$ and note any similarities between the coefficients of the $x$'s, $y$'s, and $z$'s. The goal is to reduce the system of equations in such a way that we can solve for a single variable, or, in this case, we need only reduce the system to two equations in $x$ and $z$.
I. Note that in the 1st and 2nd equation, the coefficients of the $y$-terms are, respectively, $-1$ and $1$. So simply by adding equations $[1]$ and $[2]$, we can eliminate $y$.
II. That said, it would then make the most sense to eliminate the $y$-variable in the third equation, as well; we can do so my multiplying equation $[2]$ by $3$, then adding the modified equation $[2]$ to $[3]$.
We thus end up with two equations which resolve the problem nicely. More details:
$[1] + [2] \to 5x − z = 4\qquad$ Add eq. $[1]$; the $y$-terms cancel out, leaving $5x - z = 4$:
$\; 4x-y+2z = 7$
$\underline{+ x + y - 3z = -3}$
$\;5x+0y-z =4 \implies 5x - z =4\qquad$ [I]
$3 \times [2] + [3] \to 5x-z=-9\quad$ Multiply $[2]$ by $3$: $\quad 3(x+y-3z=-3) = 3x +3y - 9z=-9$
*Note: Take this last equation [$3x +3y - 9z=-9$], and add it to equation $[3]$:
$\quad 3x +3y-9z=-9$
$\underline{+ 2x -3y +8z = a}$
$\;\;5x +0y-z=-9 +a \implies 5x -z = -9 +a\qquad$ [II]
So we're left with [I] $5x - z = 4$ and [II] $5x - z = -9 + a$.
For [I] and [II] to both hold, we must have that $4 = -9 + a$, from which we can solve for $a$ by adding 9 to both sides of the equation, giving us: $a = 13$.
The goal, using this approach, is to eliminate a variable by adding/subtracting equations (as with equation $[1]$ and $[2]$), or adding/subtracting a multiple of an equation (as we did when we multiplied equation $[2]$ by $3$, and then added to equation $[3]$).
This is essentially what row reduction will be like, except you won't have the variables cluttering everything up!