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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 7, 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. \(\displaystyle 2x=6-2\lambda\Rightarrow x=3-\lambda,\quad y=5-\lambda-(3-\lambda)=2\)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
✕\(\displaystyle x+y+z=5,\quad x-y+z=1\Rightarrow x=3+\lambda,\ y=2,\ z=\lambda\)✓\(\displaystyle x+y+z=5,\quad x-y+z=1\Rightarrow x=3-\lambda,\ y=2,\ z=\lambda\)
Substitute the free variable into the equation for x, then subtract it from 3

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, Specialist Mathematics, chapter 7: Vectors, lines and planes

In this chapter

Calculate with three-dimensional vectors and test independenceConstruct oriented cross products, normals and exact areasRecover every valid parallel-plane distance parameterRepresent and intersect spatial lines and planesSolve a linear system with three or four unknownsClassify and parameterise linear systemsDifferentiate parametric curves and integrate vector motionConstruct conic loci and convert coordinate formsUse vectors in geometry, relative motion and force models