Find intersections and classify systems

Try

Two fictional event packages use the same supplies. Package A uses two cards and one ribbon; package B uses one card and two ribbons. You have exactly 11 cards and 10 ribbons to use. Let a and b be package counts. Write both constraints before finding the counts.

The equations are 2a+b=11 and a+2b=10. Substituting b=11−2a into the second gives a+22−4a=10, so a=4 and b=3. Check both resources: 8+3=11 cards and 4+6=10 ribbons.

Predict

Predict what happens if the two constraints instead say a+2b=10 and 2a+4b=20. Do they determine one production plan? The second is a multiple of the first, so it adds no new information. Over the reals there are infinitely many pairs; whole nonnegative counts restrict the possibilities further.

Experiment

Compare three systems by graphing their lines.

Equations	Relationship	Real solution count
y=x+1 and y=−x+5	Intersect at (2,3)	One
y=x+1 and y=x+4	Parallel, distinct	None
y=x+1 and 2y=2x+2	Same line	Infinitely many

Check each classification algebraically. Equal slopes alone do not distinguish the last two rows: intercepts matter too. In elimination, these rows respectively produce a genuine equation, a contradiction or an identity.

Understand

A system solution is an ordered pair satisfying every equation at once. In substitution, replace a variable by an equal expression. In elimination, add a multiple of one equation to another so that one variable cancels. These transformations preserve the common solutions when the corresponding operation is reversible.

For 2x+y=7 and 3x−y=8, adding eliminates y and gives 5x=15. Recover x=3 and y=1, then check both originals. Reporting only x is incomplete unless the question explicitly asks for a coordinate.

A nonlinear system can produce several intersections. For y=x² and y=2x+3, set the outputs equal: x²−2x−3=0. Factor (x−3)(x+1)=0, so x=3 or −1. Recover y from either original formula, obtaining (3,9) and (−1,1). Do not combine x from one solution with y from the other.

Other configurations include tangency and no real intersection. The line y=−1 never intersects y=x² over the reals. The line y=0 meets it once at the repeated root x=0. A repeated algebraic root is still one distinct point. A graph window can miss an intersection, so use algebra to establish completeness when possible.

Context imposes another filter. A real intersection with fractional coordinates may be meaningful for amounts of liquid but impossible for counts of indivisible products. Likewise, a negative coordinate may solve an unrestricted system while violating a nonnegative inventory condition. Do not round a solution to whole counts and claim the exact constraints still hold.

Apply

The circle–line task requires both coordinate pairs and independent checks. The production task deliberately yields fractional counts: explain the distinction between solvability over the reals and feasibility as a whole-number plan.

Use the linear-systems investigation to inspect unique, parallel and coincident cases. Its line-only mechanism does not replace the nonlinear intersection reasoning in this lesson.

Continue

Polynomial division exposes factors and helps solve higher-degree equations. It also provides the algebra needed later to describe rational-function asymptotes without guessing from a graph.