Unit 4 · Core unit

Systems, Matrices, and Sequences

Three related ideas about handling many quantities at once: solving equations that must hold together, organizing coefficients into matrices, and describing a list of numbers by the rule that generates it.

What a strong answer looks like

Answers here should name the structure before the arithmetic. How many equations, how many unknowns, arithmetic or geometric, finite or infinite -- each of those decides the method.

Practice this unit Course overview

Topics in this unit

Solving Systems of Equations

Know

A solution to a system is a point that satisfies every equation at once. Two lines can meet once, never, or everywhere, and those three cases correspond exactly to one solution, none, or infinitely many.

Apply

Use elimination when a variable is easy to cancel and substitution when one equation is already solved for a variable. For three unknowns, work systematically toward one equation in one variable.

Watch out

Reaching a false statement such as 0 = 5 means no solution; reaching a true statement such as 0 = 0 means infinitely many. Students often read both as an error in their own work.

Study move

After solving, substitute your answer into every original equation, not just the one you used last.

Nonlinear Systems and Inequalities

Know

When a curve replaces one of the lines, the number of intersection points changes. A parabola and a line can meet twice, once, or not at all, and the algebra reflects that through the discriminant.

Apply

Set the expressions equal, solve the resulting equation, then substitute back to recover the second coordinate of each point.

Watch out

A solution to the combined equation gives only the x-values. Reporting them as the answer leaves the work half finished.

Study move

Sketch both curves before solving so you know how many intersection points to expect.

Matrices and Their Operations

Know

A matrix is a rectangular array used to carry the coefficients of a system. Addition works entry by entry, but multiplication pairs rows with columns, which is why the inner dimensions must agree.

Apply

Check dimensions before multiplying: an m by n times an n by p gives an m by p. Use the determinant of a 2 by 2 as the main diagonal product minus the other diagonal product.

Watch out

Matrix multiplication is not commutative. AB and BA can differ, and one can exist while the other does not.

Study move

Before computing any product, write the two dimension pairs side by side and confirm the inner numbers match.

Row Operations and Gaussian Elimination

Know

Three operations preserve the solution set of a system: swapping rows, scaling a row by a nonzero constant, and adding a multiple of one row to another. Row reduction is a disciplined way of applying them.

Apply

Work toward a leading 1 in each row and zeros below it, then back-substitute. A zero determinant signals that no unique solution exists.

Watch out

Column operations are not permitted. Columns represent variables, so swapping them silently renames the unknowns.

Study move

Narrate each row operation as you perform it. If you cannot say which of the three it was, it was probably not legal.

Sequences and Series

Know

An arithmetic sequence adds a fixed amount each step; a geometric sequence multiplies by a fixed ratio. A series is the sum of a sequence, and sigma notation is shorthand for that sum.

Apply

Use a(n) = a(1) + (n - 1)d for arithmetic terms and the partial sum formula for geometric ones. An infinite geometric series converges to a/(1 - r) exactly when the ratio is smaller than 1 in absolute value.

Watch out

The n minus 1 in the arithmetic formula is the most common slip in the unit: the first term takes no steps, so reaching the twentieth takes nineteen.

Study move

Given four terms, decide arithmetic or geometric before writing any formula, and say what the common difference or ratio is.

Emphasized in this unit

Connections and techniques that receive extra attention in this honors sequence.

• Systems and sequences as core structures for modeling several quantities or repeated change
• Matrices taught alongside systems so row operations retain their equation-level meaning
• Determinants as the quick test for whether a unique solution exists
• Infinite geometric series as the first encounter with a convergent limit

Varies by course

Related topics some schools attach to this unit and others leave out. Covered on request rather than assumed.

Partial fractions. Usually taught when the course is explicitly preparing students for integration techniques.
Mathematical induction and the Binomial Theorem. Often added in courses that treat sequences and discrete mathematics at greater depth.
Probability and combinatorics. Sometimes included here and sometimes reserved for statistics or a discrete-mathematics course.

Mastery checklist

• Can solve a linear system by elimination and by substitution.
• Can interpret an impossible or always-true result as no solution or infinitely many.
• Can find every intersection point of a line and a curve.
• Can decide whether a matrix product is defined and give its dimensions.
• Can compute a 2 by 2 determinant and say what a zero value implies.
• Can identify a sequence as arithmetic or geometric and find any term.
• Can decide whether an infinite geometric series converges and find its sum.

Check yourself

• What does a determinant of zero tell you, and what does it deliberately not tell you?
• Why must the inner dimensions match before two matrices can be multiplied?
• Why does an infinite sum of positive numbers sometimes have a finite value?

Modeling drill

A theater sells adult tickets at 12 dollars and student tickets at 8 dollars. One night it sells 250 tickets for 2,600 dollars. Set up and solve the system, then explain what the two equations represent physically and how you would check the answer is reasonable without redoing the algebra.

system of equationseliminationsubstitutionconsistentdependentmatrixdimensionsdeterminantrow operationGaussian eliminationarithmetic sequencegeometric sequencesigma notationconvergent series

Unit 3: Functions and Graphs

Unit 5: Conic Sections