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How do you solve three linear equations in three unknowns by hand?
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All Flashcards in Topic 1.16
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1.16.18 cards
How do you solve three linear equations in three unknowns by hand?
Eliminate one variable to get two equations in two unknowns, solve those, then back-substitute for the third.
How do you eliminate a variable?
Add or subtract two equations (scaling one first if needed) so that variable cancels.
How do you find the third unknown after the first two?
Back-substitute the known values into one of the original equations.
How do you solve a 3×3 system on the GDC (Paper 2)?
Use the simultaneous-equation solver (PlySmlt2) or enter the coefficient matrix and use rref.
What if the coefficients don't match to cancel?
Multiply an equation by a constant first so a variable's coefficients are equal (or opposite).
How do you check a solution (x, y, z)?
Substitute it into the equation you didn't use last; all three should hold.
Solve x + y + z = 6, x − y + z = 2, 2x + y − z = 1.
x = 1, y = 2, z = 3.
Why eliminate the SAME variable from two pairs?
It leaves two equations in the same two unknowns, which you can then solve as a 2×2 system.
1.16.28 cards
How many solutions can a system of linear equations have?
Exactly one, none, or infinitely many — never a finite number greater than one.
What does 0 = 0 at the end of elimination mean?
An equation was redundant → infinitely many solutions (planes meet in a line).
What does 0 = (non-zero) mean?
The equations contradict each other → no solution (inconsistent).
Geometrically, what is a unique solution?
The three planes meet at a single point.
Geometrically, what is 'infinitely many solutions'?
The three planes meet along a common line (or coincide).
Geometrically, what is 'no solution'?
The planes have no point common to all three (e.g. they form a triangular prism).
For a parameter k, how do you find the consistent value?
Eliminate to two copies of the same left side, then set their right-hand sides equal.
Can a linear system have exactly two solutions?
No — only 0, 1, or infinitely many.
Topic 1.16 study notes
Full notes & explanations for Systems of equations (HL only)
Math AA exam skills
Paper structures, command terms & tips
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