Systems, Counting, the Binomial Theorem + Capstones

This is Unit Nine of College Algebra: systems, counting, the binomial theorem — and the capstones that close the whole course. The new material comes first and it is genuinely new. Every unit so far handled one equation at a time. This one handles several at once, and then turns to counting — working out how many arrangements exist without producing them. Then four capstones, and one of them is worth waiting for. It goes back through every named error in the course — twenty-one of them — and sorts them into three families. Distribution, custody, and misread notation. That is a genuinely useful thing to be handed at the end, because it means a mistake you have not seen before is still diagnosable. You are not learning twenty-one separate cautions; you are learning three habits of misreading. By the end you will be able to do five things. One: solve a system by either method, and check properly. Two: say what the three outcomes of a two-line system mean. Three: tell a permutation from a combination. Four: expand a binomial power without multiplying it out. Five: name the three families every error in this course belongs to.

Five entries. The object, and two methods that do the same job by different routes — both reducing two equations in two unknowns to one equation in one unknown, which is territory from Unit Three.

The object.

Alg 481 Systems of Linear Equations

Several equations sharing variables, with a solution being one assignment making every one of them true. On a graph, the point where the lines meet.

The first method.

Alg 482 The Substitution Method

Solve one equation for a variable and put that expression into the other, which leaves a single equation in a single unknown.

Watch it run.

Alg 484 Solving by Substitution

Isolate, substitute, solve, then put the value back to get the other. And check against both equations — a pair can satisfy one of them and fail the system, which one check would not catch.

The second method.

Alg 486 The Addition Method

Scale the equations until one variable's terms are opposites, then add them so that variable disappears — leaving one unknown again.

Watch it run.

Alg 488 Solving by Addition

Multiply equations until the chosen coefficients are opposites, add to eliminate, solve, and substitute back. Scaling reaches every term in the equation — scaling half of one produces a different equation entirely.

One entry, and it explains something the procedures produce without warning: sometimes both variables vanish. That is not a failure, and what it means depends on what is left behind.

The three outcomes.

Alg 490 Independent, Dependent, Inconsistent

Two lines crossing at one point, lying on top of one another, or running parallel and never meeting — giving one solution, infinitely many, or none. The vanishing variables are reporting which case you are in.

One procedure, and its warning is the most practically useful in the unit — the equations have to count the same thing on both sides, and a mixture problem is where that goes wrong.

Watch a story become equations.

Alg 492 A System from a Story

Name the unknowns, write an equation for each fact given, solve by either method, and read the answer back in the story's own terms. Track the units through every line — a concentration equation counts the substance, not the volume holding it.

Two entries scaling the method up. Three unknowns means eliminating twice to get down to two, and there is one rule about how — break it and the work rebuilds the problem you were trying to reduce.

The larger object.

Alg 496 Systems in Three Variables

Three equations and three unknowns, with a solution being an ordered triple. The strategy reduces to a two-variable system and finishes with the methods you have.

The rule that keeps it working.

Alg 498 Solving a Three-Variable System

Pair the equations twice and eliminate the same variable in each pairing, which leaves a two-variable system to solve. Eliminating different variables in the two pairings puts back what the first elimination removed.

Two entries where a system stops being about lines. Once a curve is involved the number of meeting points is no longer fixed at one, and the substitution can produce candidates that need auditing.

The mixed case.

Alg 501 Nonlinear Systems

At least one equation graphing as a curve. A line and a parabola can cross twice, touch once, or miss entirely — so the count is part of the answer.

Solving one.

Alg 503 Solving a Nonlinear System

Substitute one equation into the other, solve the single-variable result, and pair each answer with its partner. Where a radical was involved, a quadratic step can hand back pairs that fail the original — so each pair gets checked.

Four entries on a genuinely different subject: working out how many arrangements exist without writing any of them down. The whole topic turns on one question — whether rearranging the same selection counts as a different result.

The foundation.

Alg 511 The Multiplication Principle

Choices made in sequence multiply — so many ways for the first, so many for the second, and the total is their product.

Counting orderings.

Alg 513 Factorials

Multiplying the counting numbers from a value down to one counts the ways of arranging that many distinct objects. By convention, zero of them arrange one way.

When order counts.

Alg 515 Permutations

Selecting some objects from a set and arranging them, where two selections containing the same objects in a different order are different results.

When it does not.

Alg 516 Combinations

Selecting without regard to arrangement. The formula is the previous one divided by the number of ways the chosen objects could be ordered — which removes exactly the distinction the previous one kept.

Four entries where the counting numbers turn up somewhere unexpected. The numbers counting unordered selections are exactly the coefficients appearing when a two-term expression is raised to a power — and that connection lets you write one term without expanding anything.

The reappearance.

Alg 518 Binomial Coefficients

The combination numbers, doing a second job as coefficients in an expansion — and one row of Pascal's triangle lists all of them for a given power.

The theorem.

Alg 520 The Binomial Theorem

A two-term expression raised to any power expands into a sum whose coefficients are those counting numbers, with one exponent walking down as the other climbs. Losing the middle terms is Unit Two's error at higher powers.

Watch one expand.

Alg 522 Expanding with the Binomial Theorem

Write the coefficient row, then walk the exponents in opposite directions and simplify. The exponent reaches the entire second term — a coefficient inside it gets raised too, which is where the arithmetic slips.

One term alone.

Alg 524 A Single Term of an Expansion

Any single term of the expansion can be written directly from its position, which means a question asking for one term needs none of the others.

The first of four capstones, closing this unit rather than the course — and it names what solving several equations and counting arrangements have in common, which is not obvious.

The unit's closer.

Alg 530 Many Unknowns, Counted Ways

Solving several equations at once, and counting without listing — the unit's two subjects, both running on machinery the course had already built. Every claim here carries an earlier number.

Three entries closing the whole course, and the first is the one to pay attention to. It takes every named error in nine units and shows they are not twenty-one separate things to remember.

Every error, sorted.

Alg 536 The Error Families

Twenty-one named errors falling into three families — spreading an operation where it does not reach, losing or inventing a solution, and misreading what a symbol says. Knowing the family diagnoses a mistake you have never seen before.

The course, as one thread.

Alg 538 The Course, One Thread

The whole course is one story told twice — first about numbers, then about functions — with every chapter extending the last without breaking it. That is the same claim Unit One's capstone made about its own numbers, now made about everything.

And where it stops.

Alg 540 Where the Numbers End

Nine units running from the number line to the binomial theorem, exactly as the map promised at the start. The run is built to be finished — and every claim in these closers carries an earlier number.

Five things you can now do, and College Algebra is complete. You can solve a system by substitution or by elimination, and check against every equation — a pair can satisfy one and fail the system. You can say what the three outcomes mean: lines crossing, lying on each other, or running parallel — and that vanishing variables report which case you are in. You can tell a permutation from a combination by one question — whether the same objects in a different order count as a different result. You can expand a binomial power without multiplying it out, and write a single term of the expansion without producing the others. And you can name the three families every error in this course belongs to: spreading an operation too far, losing or inventing a solution, and misreading a symbol.