3 - Three equations

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Simultaneous Equations

Section 3 Solving three simultaneous equations and three unknowns

TThe methods discussed in Sections (1) and (2) to solve systems of linear simultaneous equations with two unknowns and two equations are also applicable to solve systems with $n$ unknowns and $n$ equations. One commonly used strategy to solve a $3$ X $3$ system of linear simultaneous equations is to eliminate one of the variables first by elimination or substitution approach and then to proceed with remaining unknowns.

Example

Solve the following system of linear simultaneous equations with three unknowns and three equations: $$4x + 2y - 2z = 8 \qquad\qquad(1)$$ $$\quad\;\; x + y + z = 7 \qquad\qquad(2)$$ $$2x + 2y + z = 12 \qquad\qquad(3)$$

Multiply equation (3) by $2$ and add it with equation (1) to eliminate $z$, giving an equation with $x$ and $y$

$4x + 2y - 2z = 8 \qquad\qquad(1)$
$2 \times $ Equation (3) $\qquad\qquad 4x + 4y + 2z = 24 \qquad\qquad(4) \qquad\qquad\qquad\qquad\qquad\;\;\;$

Equation (1) plus equation (4): $$\qquad8x + 6y = 32 \qquad\qquad(5)$$ Deduct equation (2) from equation (3) to eliminate $z$ again, giving another equation with $x$ and $y$. $$\;\; 2x + 2y + z = 12 \qquad\qquad(3)$$ $$\qquad x + y + z = 7 \qquad\qquad(2)$$ Equation (3) minus equation (2): $$\qquad\qquad x + y = 5 \qquad\qquad(7)$$ Now, from equations (5) and (7) we get: $$8x + 6(5 - x) = 32 \qquad\qquad$$ $$\qquad\qquad\quad\; 2x = 2 \qquad\qquad$$ $$\qquad\qquad\qquad x = 1 \qquad\qquad$$ As $x=1$, from $x+y=5$, we get $y=4$ and from $x+y+z=7$, we get $z=2$.