Solve a multi-equation system
Reduce a system by eliminating the same variable from two independent pairs. Solve the smaller system, recover the omitted value and check the result in an original equation
\[a_1x+b_1y+c_1z=d_1\]
- 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 \(\displaystyle z=3,\quad y=2,\quad x=1\)
Worked example
Solve the system below
\[\begin{aligned}R_1&:x+y+z=6\\R_2&:2x-y+z=3\\R_3&:x+2y-z=2\end{aligned}\]
- \(\displaystyle R_2-2R_1:-3y-z=-9,\quad R_3-R_1:y-2z=-4\)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.
Common mistakes
Not removing x from both requested reductions
✕\(\displaystyle R_2-2R_1:-3y-z=-9,\quad R_3-2R_1:-x-3z=-10\)✓\(\displaystyle R_2-2R_1:-3y-z=-9,\quad R_3-R_1:y-2z=-4\)
Both displayed equalities are valid, but the second still contains x. For the requested two-variable subsystem in y and z, use R3 − R1 instead. Other full-system methods may use different intermediate reductionsChanging a sign in a reduced equation
✕\(\displaystyle R_2-2R_1:3y-z=-9\)✓\(\displaystyle R_2-2R_1:-3y-z=-9\)
The row operation must be applied to every coefficient and the constant with its correct signStopping before recovering the final variable
✕\(\displaystyle z=3,\quad y=2\)✓\(\displaystyle z=3,\quad y=2,\quad x=1\)
A three-unknown system is not complete until the omitted variable has been recoveredDifficulty levels
introSolve a three-unknown system with one substitution-ready equation
standardSolve a three-unknown system by eliminating the same variable from two pairs
challengeUse a dense or mixed four-unknown system with technology support and exact verification
examSelect a valid reduction and enter the complete solution vector for a four-unknown system
Questions about this topic
Why eliminate the same variable from both pairs?+
This gives two equations in the same two remaining unknowns, ready to solve directly. Other valid row operations are possible, but may need another elimination step
What is a pivot equation used for?+
It is used to eliminate one variable from the other equations by substitution or row operations
Why is the original system checked at the end?+
Substitution into an original equation checks that the recovered values still satisfy the given system
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 1: Solving equations
In this chapter
Factorise, solve and reconstruct a rational quadraticSolve a quadratic exactly with the quadratic formulaSolve an equation that reduces to quadratic formDetermine parameter values from a discriminant conditionDivide polynomials and state quotient and remainderApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsDetermine parameters in a polynomial identityUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityUse exact trigonometric identities and quadrant informationSolve circular equations over finite domainsGeneral solutions of circular equations and interval extractionLocate and bisect a circular-equation rootRearrange literal equations and recognise equivalent line formsSolve a pair of simultaneous linear equationsSolve a linear system with three or four unknownsVerify coordinate incidence and determine a parameterSolve literal equations and simultaneous literal equationsClassify and parameterise linear systems