Systems of Equations & Problem-Solving Vocabulary

60 vocabulary flashcards summarizing key terms, strategies, and concepts from the lecture on units, linear equations, systems of equations, and general mathematical problem-solving techniques.

Units

The measurement labels attached to quantities; keeping them consistent prevents calculation errors.

System of Equations

A set of two or more equations considered simultaneously to find common solutions.

Linear Equation

An equation whose highest degree on every variable is one.

Variable

A symbol—commonly x, y, or z—used to represent an unknown quantity.

Substitution Method

Solving a system by isolating one variable in an equation and replacing it in the other equation(s).

Elimination Method

Solving a system by adding or subtracting equations to cancel one variable.

Graphical Method

Plotting equations on a coordinate plane and locating their point of intersection.

Point of Intersection

The point where graphs of equations meet; represents the simultaneous solution.

Degrees of Freedom

The number of values that can vary independently, often calculated as n – 1 in statistics.

n-1 Rule

The adjustment used when computing sample variance or standard deviation: divide by (n – 1).

Coefficient

The numerical factor multiplied by a variable in an algebraic term.

Constant Term

A number in an equation that has no variable attached.

Highest Degree

The largest exponent present in an equation; determines linearity or non-linearity.

Simplify

The process of reducing an expression or equation to its most basic form.

Formulate

To translate a word problem into algebraic equations.

Critical Thinking

Evaluating and connecting ideas logically while solving mathematical problems.

Begin with an End in Mind

Problem-solving attitude of identifying the desired answer before selecting methods.

Principle over Formula

Understanding underlying concepts instead of relying solely on memorized formulas.

Least Common Multiple (LCM)

The smallest number that is a multiple of two or more given numbers; used to align coefficients.

Sample vs Population

A sample is a subset of observations; the population is the entire set.

Standard Deviation

A measure of data spread; square root of variance.

Variance

The average of squared deviations from the mean; shows variability.

Equation

A mathematical statement indicating two expressions are equal.

Unknown

A quantity in an equation whose value must be determined.

Representing Quantities with Variables

Assigning symbols to real-world amounts to create equations.

Dollar-Value Equation

An equation where coefficients reflect monetary amounts (e.g., $2.20 x + $3 y).

Gallons Problem

Typical word problem involving gasoline types, prices, and total sales.

Two-Equation System

The simplest system containing exactly two linear equations in two variables.

Consistent System

A system that has at least one solution.

Inconsistent System

A system with no possible solution; equations contradict each other.

Solution Set

All ordered pairs (or triples) that satisfy every equation in a system.

Check / Verify Solution

Substituting found values back into original equations to confirm correctness.

Mental Math

Performing calculations without external aids; speeds up checking steps.

Practice Sets

Collections of problems designed for repetitive skill reinforcement.

Algebrahomework.com

An online tool mentioned for daily algebra practice and problem solving.

Excess and Deficiency Method

Guess-and-adjust strategy for integer word problems (e.g., tickets or bills).

Word Problem

A real-life scenario that must be modeled mathematically to find an answer.

Translating Words to Equations

Converting verbal descriptions into algebraic expressions and equalities.

Feet-and-Legs Problem

Classic system involving animals or objects with different numbers of legs.

Ticket Sales Problem

System problem where adult and child ticket counts must meet both total and revenue conditions.

Cashier Bills Problem

System problem involving counts of $5 and $10 bills totaling specified numbers and values.

Degree of an Equation

The highest power on any variable within the equation.

Nonlinear Equation

An equation where the highest degree is greater than one.

Quadratic Equation

A second-degree polynomial equation, generally ax² + bx + c = 0.

Homogeneous System

A system where all constant terms are zero; always includes the trivial solution.

Isolate a Variable

Manipulating an equation so that a chosen variable stands alone on one side.

Substitute Back

Inserting the value of one variable into another equation to find remaining unknowns.

Additive Inverse

The number that, when added to a given number, yields zero (e.g., –a is inverse of a).

Multiplicative Inverse

The number that, when multiplied by a given number, yields one (e.g., 1/a is inverse of a).

Divide Both Sides

Operation that keeps an equation balanced by dividing every term by the same nonzero value.

Multiply Both Sides

Operation that maintains equality by multiplying all terms by the same factor.

Common Sense Check

A quick plausibility test to see if numeric answers make real-world sense.

Units Consistency

Ensuring both sides of an equation carry the same units throughout calculations.

Problem-Solving Steps

Typical sequence: understand, formulate, simplify, solve, and verify.

Population Size (N)

Total number of observations or individuals in an entire group.

Sample Size (n)

Number of observations selected from the population.

Square Root Extraction

Operation used to solve for a variable when it appears squared.

Positive and Negative Roots

The two possible square roots of a positive number (e.g., ±2 for 4).

Practice, Practice, Practice

Repeated application of concepts to reinforce understanding and speed.

Online Learning Tools

Websites, videos, or software that support independent study and drill.