When do equations identify one point?
How can row subtraction reveal one, no or infinitely many solutions?
Entering midway? An equation is a condition on unknown values. Subtracting equal quantities preserves equality.
Start with a concrete question
Compare x+y=2 with a second equation, and inspect both row reduction and geometry.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict before calculating: How can row subtraction reveal one, no or infinitely many solutions?
Change one thing. Inspect what happens.
- Solve the initial pair by subtraction and substitution.
- Duplicate the first row and explain the free variable.
- Choose conflicting parallel rows; identify the contradiction.
- State rank as a count of independent constraints, not a count of written rows.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Two equations on fixed equal scales
Horizontal: x. Vertical: y.
- 1. x+y=2 (solid line)
- 2. x−y=4 (dashed line)
- 3. unique solution (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | x | y |
|---|---|---|
| x+y=2 | -4 | 6 |
| x+y=2 | 6 | -4 |
| x−y=4 | -2 | -6 |
| x−y=4 | 6 | 2 |
| unique solution | 3 | -1 |
Follow the calculation
- Row 1: x + y = 2.
- Row 2: x − y = 4.
- Subtract row 1 from row 2: -2y = 2.
- y=-1; substitute in row 1: x=2−-1=3.
Independent equations identify one point.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
When do equations identify one point?
Compare x+y=2 with a second equation, and inspect both row reduction and geometry.
Concepts to recall
An equation is a condition on unknown values. Subtracting equal quantities preserves equality.
Understand the mechanism
x+y=2 and x−y=4 imply −2y=2, so y=−1 and x=3.
A duplicated row adds no information: x+y=2 still allows every point on one line.
If the same left side must equal both 2 and 4, no pair can satisfy both. Coefficient rank counts independent left sides; augmented rank also accounts for the right side.
Symbols and calculation
Subtract row 1 from row 2; a nonzero pivot determines a variable, 0=0 leaves freedom, and 0=c with c≠0 is inconsistent.
Worked example
Row 1: x + y = 2.
Row 2: x − y = 4.
Subtract row 1 from row 2: -2y = 2.
y=-1; substitute in row 1: x=2−-1=3.
Your investigation
Solve the initial pair by subtraction and substitution.
Duplicate the first row and explain the free variable.
Choose conflicting parallel rows; identify the contradiction.
State rank as a count of independent constraints, not a count of written rows.
Limits and interpretation
- These are bounded teaching models with explicit assumptions, not evidence of a real-world intervention.
Try a fresh case
Solve x+y=7 and x−y=1. What is x?
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Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Explain a fresh small numerical case with a diagram or table and an argument.
College investigation
Implement the method, test a boundary or counterexample, and justify its assumptions.
Workplace investigation
Use a small decision scenario, state units and constraints, and test how the recommendation changes.
Keep the evidence, choose your next step
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