Race to the Intersection

Every line is a racer. Its slope is its speed, its y-intercept is its head start — and the intersection is the moment one racer catches the other. Build the race, read the graph, and let algebra call the photo finish.

Grade 8 · Unit 3 8.EE.5 Slope as unit rate 8.EE.6 Similar triangles → y = mx + b 8.EE.7 Linear equations 8.EE.8 Systems
Part 1

Build the Race

Set each racer's speed (slope) and head start (y-intercept). The graph shows distance over time; the equations update live. Watch what happens to the meeting point when the speeds match — and when everything matches.

🟢 Racer A

4
0
y = 4x

🟡 Racer B

2
6
y = 2x + 6
Part 2

Photo Finish

Five races, called before they run. Lock in your prediction first — then the reveal does the algebra and shows exactly where (or whether) the racers meet.

Race 1 of 5
Part 3

The Line Machines

Three machines behind the racetrack: measure any line's slope with triangles, solve any one-variable equation (and classify its solutions), and crack any system.

Slope is rise over run — and thanks to similar triangles, ANY two points on a line give the same answer.

Combine, distribute, undo — and sometimes discover the equation has no solution, or is true for every x.

Two equations, two unknowns, one point that satisfies both. Substitute, solve, check — in the graph AND the algebra.

Part 4

Finish-Line Quiz

Ten questions from the starting gun to the photo finish. Wrong answers get a hint naming the exact machine or race that proves the right move.

📦 Word Box

Slope (m)
Rise over run — how much y changes for each step of x. In a race graph, it's the racer's speed.
Unit rate
The amount per ONE unit — meters per one second. On a proportional graph, the unit rate IS the slope.
y-intercept (b)
Where the line crosses the y-axis — the racer's head start at time zero.
Rate of change
Another name for slope: the constant amount added with every step.
Proportional relationship
A line through the origin, y = mx — no head start, distance always speed × time.
Linear equation
An equation whose graph is a straight line, writable as y = mx + b.
System of equations
Two (or more) equations sharing the same variables — two racers on the same track.
Solution of a system
The point that makes BOTH equations true — the intersection where the racers meet.
Parallel lines
Same slope, different intercepts. Same speed, different head starts: the gap never closes — no solution.
No solution
The equation or system collapses to something false (like 5 = 9). The racers never meet.
Infinitely many solutions
The equation collapses to something always true (0 = 0) — the two racers are the same line, tied at every moment.
Distributive property
a(b + c) = ab + ac. Distribute BEFORE combining — 2(x + 1) is 2x + 2, never 2x + 1.
Like terms
Terms with the same variable part: 3x and 2x combine to 5x; 4 and 5 combine to 9.