Satisfy two rules at once
When do two linear requirements agree?
Entering midway? A line y=mx+b has slope m and intercept b. A shared point must satisfy both equations.
Start with a concrete question
Two original algebraic rules describe the same pair x,y. Change slopes and intercepts to explore unique, absent and infinitely many intersections.
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 whether equal slopes always mean the same line.
Change one thing. Inspect what happens.
- At defaults equate the two y expressions and find x, then y.
- Make slopes equal but intercepts different.
- Make both slopes and intercepts equal; explain why reporting one intersection would be misleading.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Two simultaneous constraints
Horizontal: x. Vertical: y.
- 1. First equation (solid line)
- 2. Second equation (dashed line)
Axes use the displayed ranges.
Read plotted values as a table
| Series | x | y |
|---|---|---|
| First equation | -10 | -9 |
| First equation | 10 | 11 |
| Second equation | -10 | 13 |
| Second equation | 10 | -7 |
Follow the calculation
- First rule y = 1x + 1; second rule y = -1x + 3.
- At intersection: (1 − -1)x = 3 − 1.
- x = 1; substitute into first rule: y = 2.
Solving a system means satisfying both rules at once. Equal slopes do not always mean no solution: identical lines are different from parallel distinct lines.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Satisfy two rules at once
Recap for entering here
A line y=mx+b has slope m and intercept b. A shared point must satisfy both equations.
The concrete question
Two original algebraic rules describe the same pair x,y. Change slopes and intercepts to explore unique, absent and infinitely many intersections.
Predict whether equal slopes always mean the same line.
Work through the mechanism
Substitution replaces one expression with an equal expression. At an intersection ax+b=cx+d, so (a−c)x=d−b.
For y=x+1 and y=−x+3, 2x=2 gives x=1 and y=2.
If slopes agree, the x term cancels. Remaining constants either disagree (no solution) or agree (every point on the same line).
Formula and symbols
For a≠c: x=(d−b)/(a−c), then y=ax+b. a,c are slopes; b,d are intercepts.
Test the idea
At defaults equate the two y expressions and find x, then y.
Make slopes equal but intercepts different.
Make both slopes and intercepts equal; explain why reporting one intersection would be misleading.
Assumptions and boundaries
Only two linear equations are modeled; nonlinear systems can have multiple isolated intersections.
The display window may omit an intersection even when the numerical solution exists.
Extend your investigation
High school: Compare two fictional fee plans and find their break-even usage.
College: Solve a three-variable system by elimination and verify every equation.
Working professionals: Document a capacity model with simultaneous resource constraints.
Try a fresh case
For y=2x+1 and y=−x+7, what is the intersection x?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Compare two fictional fee plans and find their break-even usage.
College investigation
Solve a three-variable system by elimination and verify every equation.
Workplace investigation
Document a capacity model with simultaneous resource constraints.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.