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Grade 8 · Linear systems

Motion tracker: Locate simultaneous solutions

Motion tracker investigation: locate simultaneous solutions. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Motion tracker: Locate simultaneous solutions. System classification: One intersection. Intersection: (0.286, 0.571). Moving probe x: -8Teal: y=2x · amber: y=-1.5x+(1)-8-4048-40040xNavy point: the common solution
Both equations describe real lines. Parallel and coincident cases are supported explicitly. The fixed plot window is x∈[−8,8], y∈[−40,40]; an intersection outside it is still reported numerically.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

System classification
One intersection
Intersection
(0.286, 0.571)
Moving probe x
-8

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

A shared solution satisfies two line equations at the same input and output. Different slopes yield one crossing, equal slopes with different intercepts yield none, and identical rules share infinitely many points. Two equations alone do not guarantee a unique solution. This investigation begins with first slope a = 2; second slope b = -1.5; second intercept c = 1. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

y = ax and y = bx+c; (a−b)x = c

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Read the two rules separately, including the second rule's intercept c.

  2. STEP 2

    Follow the changing model

    Playback moves a vertical probe across both graphs so their outputs can be compared at one input.

  3. STEP 3

    Check and explain the relationship

    If slopes differ, solve c/(a−b) and substitute into both rules. If slopes match, compare intercepts instead of dividing by zero.

Your turn to explain

Make a prediction. Test your reasoning.

Motion tracker: Solve y=2x and y=-1.5x+(1).

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

(0.286, 0.571). Both rules have output 0.571 at x=0.286.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 8 · Algebra

Solve two linked conditions by substitution

A system's solution must satisfy both equations at once; substitution replaces a variable by an equal expression.