Algebra

The One Thing

Algebra provides a systematic language for representing quantities and relationships, then solving for unknowns under specified conditions. Reliable applied solutions require correct algebraic operations, attention to domains and restrictions, and interpretation of the result in its original context.

Definitions and Results

• Variable: A symbol, usually a letter, representing an unknown or changing quantity.
• Constant: A fixed value that does not change in a given situation.
• Algebraic Expression: A combination of numbers, variables, and operations, such as $3x+5$.
• Polynomial: An algebraic expression made from variables with non-negative integer powers and numerical coefficients.
• Linear Equation: An equation in which the highest power of the variable is $1$, such as $2x+3=9$.
• Quadratic Equation: An equation of the form $a{x}^2+bx+c=0$, where $a\ne 0$.
• Arithmetic Progression: A sequence in which the difference between consecutive terms remains constant.
• Geometric Progression: A sequence in which the ratio between consecutive non-zero terms remains constant.
• Sequence: An ordered list of numbers or terms following a definite pattern.
• Series: The sum of the terms of a sequence.
• Function: A rule that assigns exactly one output to each permitted input.
• Domain: The set of allowed input values of a function.
• Range: The set of output values produced by a function.
• Logarithm: The exponent to which a positive base must be raised to obtain a given positive number.
• Matrix: A rectangular arrangement of numbers or algebraic quantities in rows and columns.
• Determinant: A scalar value calculated from a square matrix, useful for testing solvability and finding matrix inverses.
• System of Equations: Two or more equations considered together, with common values satisfying all of them.
• Inequality: A mathematical statement comparing quantities using symbols such as $<$, $>$, $\le$, or $\ge$.
• Identity: An equation that is true for every permitted value of its variable, such as

$$(a+b{)}^2=a^2+2ab+b^2.$$

• Linear-equation result: For $ax+b=0$, where $a\ne 0$,

$$x=-\frac{b}{a}.$$

• Quadratic formula: For $a{x}^2+bx+c=0$, where $a\ne 0$,

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Discriminant: The discriminant is

$$D=b^2-4ac.$$ If $D>0$, there are two distinct real roots; if $D=0$, there is one repeated real root; if $D<0$, there are no real roots.

• Arithmetic-progression nth term: If the first term is $a$, the common difference is $d$, and the nth term is $T_n$,

$$T_n=a+(n-1)d.$$

• Arithmetic-progression sum: The sum of the first $n$ terms is

$$S_n=\frac{n}{2}[2a+(n-1)d]$$ or $$S_n=\frac{n}{2}(a+l),$$ where $l$ is the last term.

• Geometric-progression nth term: If the first term is $a$ and the common ratio is $r$,

$$T_n=a{r}^{n-1}.$$

• Finite geometric-progression sum: When $r\ne 1$,

$$S_n=\frac{a(r^n-1)}{r-1},$$ equivalently, $$S_n=\frac{a(1-r^n)}{1-r}.$$

• Infinite geometric-progression sum: When $|r|<1$,

$$S_{\infty }=\frac{a}{1-r}.$$

• Function notation: For a function $f$, $f(x)$ represents the output corresponding to input $x$.
• Equality-preserving operations: When solving equations, the same operation must be applied to both sides.
• Inequality rule: Multiplying or dividing an inequality by a negative number reverses the inequality sign.
• Matrix multiplication condition: The product of two matrices is defined only when the number of columns in the first matrix equals the number of rows in the second.
• $2\times 2$ determinant: For

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],|A|=ad-bc.$$

• Matrix inverse condition: A square matrix has an inverse only when its determinant is non-zero.
• System-solvability condition: For a system of two linear equations, a unique solution generally exists when the determinant of the coefficient matrix is non-zero.
• Domain restrictions: Denominators must be non-zero. For real-valued expressions involving even roots, the quantity inside the root must be non-negative.
• Applied-model condition: Algebraic models should include meaningful units, realistic restrictions, and answers that fit the original situation.

Worked Methods

Solving a linear equation

• Represent the unknown quantity with a variable.
• Apply the same operation to both sides of the equation.
• Isolate the variable.
• For

$$ax+b=0,a\ne 0,$$ subtract $b$ and divide by $a$: $$ax=-b,x=-\frac{b}{a}.$$

• Check the result in the original equation and interpret it using appropriate units or restrictions.

Solving a quadratic equation

• Write the equation in the form

$$a{x}^2+bx+c=0,a\ne 0.$$

• Identify $a$, $b$, and $c$.
• Calculate the discriminant:

$$D=b^2-4ac.$$

• Substitute into the quadratic formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Use the discriminant to determine the nature of the roots:

- $D>0$: two distinct real roots; - $D=0$: one repeated real root; - $D<0$: no real roots.

• Check the roots in the original equation and reject values that violate the original context.

Finding an arithmetic-progression term or sum

• Identify the first term $a$, common difference $d$, number of terms $n$, and, where relevant, last term $l$.
• For the nth term, use

$$T_n=a+(n-1)d.$$

• For the sum of the first $n$ terms, use either

$$S_n=\frac{n}{2}[2a+(n-1)d]$$ or, when the last term is known, $$S_n=\frac{n}{2}(a+l).$$

• Interpret the result in the applied setting, such as repeated payments, production, or another regular change.

Finding a geometric-progression term or sum

• Identify the first term $a$, common ratio $r$, and number of terms $n$.
• Find the nth term using

$$T_n=a{r}^{n-1}.$$

• For a finite sum, provided $r\ne 1$, use

$$S_n=\frac{a(r^n-1)}{r-1}$$ or $$S_n=\frac{a(1-r^n)}{1-r}.$$

• For an infinite sum, first verify the condition $|r|<1$, then use

$$S_{\infty }=\frac{a}{1-r}.$$

• Relate the result to applications such as growth, depreciation, savings, or repeated changes.

Working with functions

• Identify the rule defining the function.
• Determine the permitted input values, which form the domain.
• Substitute an allowed input $x$ into the rule.
• Evaluate the resulting output $f(x)$.
• Describe the range as the set of outputs produced by the permitted inputs.
• Check that each permitted input is assigned exactly one output.

Multiplying matrices and using determinants

• Check that the number of columns in the first matrix equals the number of rows in the second.
• Multiply each row of the first matrix by each column of the second and add the corresponding products.
• For a $2\times 2$ matrix

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$ calculate $$|A|=ad-bc.$$

• Use the determinant to assess solvability and invertibility:

- if $|A|\ne 0$, the square matrix has an inverse; - for a system of two linear equations, a non-zero determinant of the coefficient matrix generally indicates a unique solution.

• Retain the dimensions, units, and interpretation of the original system.

Solving inequalities

• Perform equivalent operations on both sides.
• Isolate the variable.
• If multiplying or dividing by a negative number, reverse the inequality sign.
• State the resulting set of permitted values.
• Check the result against any domain or contextual restrictions.

Constructing and checking an algebraic model

• Define variables for unknown or changing quantities.
• Translate the relationships in the situation into expressions, equations, functions, sequences, or systems of equations.
• Solve using valid algebraic operations.
• Respect restrictions, including non-zero denominators and non-negative radicands for real even roots.
• Check the result in the original equation or context.
• Report meaningful units and reject answers that are unrealistic or incompatible with the situation.

Where It Goes Wrong

• Applying an operation to only one side of an equation instead of applying the same operation to both sides.
• Failing to reverse an inequality sign after multiplying or dividing by a negative number.
• Using the quadratic formula without first identifying $a$, $b$, and $c$, or misinterpreting the discriminant $D=b^2-4ac$.
• Using arithmetic-progression formulas for a geometric progression, or omitting the condition $|r|<1$ when finding an infinite geometric sum.
• Multiplying matrices when the number of columns in the first matrix does not equal the number of rows in the second.
• Forgetting that denominators must be non-zero, that real even roots require non-negative quantities inside the root, or that an applied answer must satisfy the original units, restrictions, and context.

What Gets Asked

This material supports questions requiring students to:

• Define and distinguish variables, constants, expressions, polynomials, equations, inequalities, identities, sequences, series, functions, matrices, determinants, and systems of equations.
• Solve linear equations and quadratic equations using the linear formula and quadratic formula.
• Determine the nature and number of quadratic roots from the discriminant.
• Find terms and sums in arithmetic and geometric progressions, including infinite geometric sums.
• Model repeated payments, growth, depreciation, savings, production, rates, costs, and measurements using algebraic expressions, equations, functions, sequences, and series.
• Evaluate functions and identify their domains and ranges.
• Multiply matrices, calculate $2\times 2$ determinants, and use determinants to assess matrix inverses and systems of equations.
• Solve inequalities while applying the correct sign-reversal rule.
• Identify domain restrictions involving denominators and even roots.
• Check algebraic solutions in the original equation or applied context and report appropriate units and realistic constraints.