Three and four unknowns
Define the unknowns and record each original condition. For a manual three-variable reduction, remove the same variable to obtain a two-variable subsystem, solve it, recover the missing value, then check all originals. Four-variable technology still requires correct inputs, exact labelled output and checks
- Keep the variable labels Record every value beside the variable it belongs to \(\displaystyle (x,y,z)\)
- Eliminate the same variable Use the same pivot variable when forming both reduced equations \(\displaystyle R_2-2R_1,\quad R_3-R_1\)
- Solve the reduced pair Use the two reduced equations to find the variables that remain \(\displaystyle -3y-z=-9,\quad y-2z=-4\)
- Recover and check Find the variable removed first and verify the values in an original equation \(\displaystyle z=3,\quad y=2,\quad x=1\)
- Translate before solving Use consistent labels and units in every condition; keep a zero for an absent variable when using a coefficient table \(\displaystyle 2c+f+0b+h=11\)
- Four-variable technology Select four unknowns, declare their order, enter every original equation and request exact output where supported. Record all four labels and check every original condition
- Count independent conditions Repeated equations do not supply new information. The practice bank uses non-singular systems, but arbitrary systems may have no solution or many solutions
Worked example
Solve the system x + y + z = 6, 2x - y + z = 3 and x + 2y - z = 2. The equations are labelled R_1, R_2 and R_3 in the order shown
- \(\displaystyle R_2-2R_1:-3y-z=-9,\quad R_3-R_1:y-2z=-4\)Eliminate x from both equations using the first equation
- \(\displaystyle y=2z-4\Rightarrow-3\left(2z-4\right)-z=-9\Rightarrow z=3\)Use the two reduced equations to solve for z.
- \(\displaystyle y-2\left(3\right)=-4\Rightarrow y=2\)Substitute z = 3 into the second reduced equation
- \(\displaystyle x+2+3=6\Rightarrow x=1\quad\Rightarrow\quad(x,y,z)=(1,2,3)\)Substitute y = 2 and z = 3 into the first equation to recover x.
- \(\displaystyle \begin{gathered}1+2+3=6\\2(1)-2+3=3\\1+2(2)-3=2\end{gathered}\)Substitute the complete triple into every original equation. All three left sides match their right sides
Common mistakes
Difficulty levels
Questions about this topic
Must every valid method eliminate the same variable?+
No. For the two-equation subsystem taught here, removing the same variable makes the remaining unknown set match. Other valid full-system methods can use different intermediate reductions; a valid equality is not wrong merely because it uses another method
Do four equations always determine four unknowns?+
No. Enough independent information is required, and the equations must be consistent. This practice bank deliberately supplies systems with a unique solution
What does the final response assess?+
One complete labelled vector is one joint solving result, even though its cells earn partial marks. It does not independently demonstrate written elimination, unrestricted model formulation, calculator operation, separate arithmetic, checking or absence of assistance
Must I use a particular calculator command?+
No device-specific syntax is assumed here. Use your supported simultaneous-equation tool or an exact algebraic method, keeping the same equations and variable labels. A supplied technology result in the lesson is an interpretation task, not proof that you operated a tool
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 1: Linear relations and coordinate geometry