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Classify and parameterise linear systems

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Classify and parameterise linear systems

A parameter can change two lines from intersecting once to being parallel or coincident. Each equation must define a line: its x- and y-coefficients cannot both be zero. Calculate the determinant first; if it is zero, scale one whole equation before comparing constants

\[D=a_{1}b_{2}-a_{2}b_{1}\]
  1. 1Write comparable equations Use the same variable order in both equations before comparing their coefficients \(\displaystyle a_{1}x+b_{1}y=c_{1},\quad a_{2}x+b_{2}y=c_{2}\)
  2. 2Use the determinant for uniqueness A non-zero determinant means the two coefficient rows are not proportional \(\displaystyle D\ne0\Rightarrow\text{one solution}\)
  3. 3Compare scaled constants when D is zero Scale one coefficient row to match the other, then compare the constants after applying the same non-zero scale factor k \(\displaystyle a_{1}=ka_{2},\quad b_{1}=kb_{2},\quad c_{1}\ne kc_{2}\Rightarrow\text{no solution}\)
  4. 4Recognise coincident lines If the same non-zero factor matches both coefficients and constants, the equations describe the same line and infinitely many solutions \(\displaystyle 2x+4y=6,\quad x+2y=3\Rightarrow\text{infinitely many solutions}\)
Year 12, VCE Units 3 & 4, Chapter 1, 4 difficulty levels, No calculatorBuild a worksheet

Classify and parameterise linear systems

A parameter can change two lines from intersecting once to being parallel or coincident. Each equation must define a line: its x- and y-coefficients cannot both be zero. Calculate the determinant first; if it is zero, scale one whole equation before comparing constants

\[D=a_{1}b_{2}-a_{2}b_{1}\]
  1. Write comparable equations Use the same variable order in both equations before comparing their coefficients \(\displaystyle a_{1}x+b_{1}y=c_{1},\quad a_{2}x+b_{2}y=c_{2}\)
  2. Use the determinant for uniqueness A non-zero determinant means the two coefficient rows are not proportional \(\displaystyle D\ne0\Rightarrow\text{one solution}\)
  3. Compare scaled constants when D is zero Scale one coefficient row to match the other, then compare the constants after applying the same non-zero scale factor k. \(\displaystyle a_{1}=ka_{2},\quad b_{1}=kb_{2},\quad c_{1}\ne kc_{2}\Rightarrow\text{no solution}\)
  4. Recognise coincident lines If the same non-zero factor matches both coefficients and constants, the equations describe the same line and infinitely many solutions \(\displaystyle 2x+4y=6,\quad x+2y=3\Rightarrow\text{infinitely many solutions}\)

Worked example

Find the solution set of the system in terms of λ, where λ is real

\[x+y+z=5,\quad x-y+z=1\]
  1. \(\displaystyle z=\lambda\)Choose z as the free variable
  2. \(\displaystyle x+y=5-\lambda,\quad x-y=1-\lambda\)Substitute z = λ and move the parameter terms to the right
  3. \(\displaystyle 2x=6-2\lambda\Rightarrow x=3-\lambda,\quad y=2\)Add the two equations to find x, then substitute into either equation to find y.
  4. \(\displaystyle x=3-\lambda,\quad y=2,\quad z=\lambda,\quad \lambda\in\mathbb{R}\)One free real parameter gives the complete solution set

Common mistakes

Comparing gradients only
✕\(\displaystyle 2x+4y=6,\quad x+2y=3\Rightarrow\text{parallel}\)✓\(\displaystyle 2x+4y=6,\quad x+2y=3\Rightarrow\text{coincident}\)
Matching gradients only is not enough; the constants must also be compared
Calling parallel lines coincident
✕\(\displaystyle 2x+4y=6,\quad x+2y=5\Rightarrow\text{infinitely many solutions}\)✓\(\displaystyle 2x+4y=6,\quad x+2y=5\Rightarrow\text{no solution}\)
After multiplying the second equation by 2, its coefficients match the first but its constant is 10, not 6. The lines are parallel, not coincident
Changing the sign in back-substitution
✕\(\displaystyle x+y+z=5,\quad x-y+z=1\Rightarrow x=3+\lambda,\ y=2,\ z=\lambda\)✓\(\displaystyle x+y+z=5,\quad x-y+z=1\Rightarrow x=3-\lambda,\ y=2,\ z=\lambda\)
The sum of x and z is three, so substituting λ for z gives three minus λ for x

Difficulty levels

introClassify a two-equation system at a stated parameter value
standardFind the exceptional parameter value and identify the parallel or coincident case
challengeExpress a two-equation, three-variable solution set with a parameter or classify a two-parameter system
examSelect a correct parameterised solution form and fill one labelled coefficient

Questions about this topic

What does a zero determinant tell us?+

For two equations that each define a line, it means the coefficient rows are proportional. Scale one whole equation to match the other coefficient row. Equal constants then give coincident lines; unequal constants give parallel lines

What is the difference between parallel and coincident lines?+

Distinct parallel lines have no common solution: after their coefficient rows are scaled to match, their constants differ. Coincident lines match after scaling the whole equation and share every point

Why does a parameterised solution use a free variable?+

For a consistent system of two independent equations in three variables, one variable is free. Set it to λ and solve for the others; all real λ then describe the solution line

Can I use a calculator?+

No. This topic is practised without a 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