Glossary
Age Problems
Word problems that involve finding the current or future ages of individuals based on given relationships between their ages.
Example:
A problem asking 'If John is twice as old as Mary now, and in 5 years he will be 10 years older than her, how old are they?' is an age problem.
Coefficient
A numerical factor multiplied by a variable in an algebraic term.
Example:
In the expression 7y, the number 7 is the coefficient of y.
Combine Like Terms
The process of simplifying an algebraic expression by adding or subtracting terms that have the same variable raised to the same power.
Example:
To simplify 3x + 5 + 2x, you would combine like terms to get 5x + 5.
Constant
A fixed numerical value in an equation or expression that does not change.
Example:
In the equation y = 3x + 10, the number 10 is a constant representing a fixed starting amount.
Initial Value
The starting amount or quantity of a dependent variable when the independent variable is zero, often represented by the y-intercept.
Example:
If a savings account starts with 500 is the initial value before any deposits or withdrawals.
Inverse Operations
Operations that undo each other, used to solve equations by isolating variables.
Example:
Addition is the inverse operation of subtraction, and multiplication is the inverse of division.
Isolate the Variable
The process of manipulating an equation to get the variable by itself on one side of the equals sign.
Example:
To solve 4x - 7 = 13, you need to isolate the variable x by adding 7 to both sides, then dividing by 4.
Linear Equations
Equations that represent a straight line, showing a relationship where quantities change at a constant rate.
Example:
Calculating the total cost of a taxi ride based on a fixed fee and a per-mile charge involves a linear equation.
Mixture Problems
Word problems that involve combining two or more quantities with different concentrations or values to form a new mixture.
Example:
Determining how much of a 10% salt solution to mix with a 30% salt solution to get a 25% solution is a mixture problem.
Motion Problems
Word problems that involve distance, rate (speed), and time, often using the formula `distance = rate × time`.
Example:
Calculating how far a car travels if it drives at 60 mph for 3 hours is a motion problem.
Rate of Change
How much one quantity changes in relation to another quantity, often represented by the slope in linear relationships.
Example:
If a plant grows 2 inches per week, its rate of change in height is 2 inches/week.
Slope
A measure of the steepness and direction of a line, representing the rate of change between two variables.
Example:
In a graph showing distance traveled over time, the slope of the line represents the speed.
Slope-intercept form
A common way to write linear equations, `y = mx + b`, where `m` is the slope and `b` is the y-intercept.
Example:
The equation y = 2x + 5 is in slope-intercept form, indicating a slope of 2 and a y-intercept of 5.
Variable
A symbol, typically a letter, representing an unknown quantity that can change or vary.
Example:
In the equation 2x + 5 = 11, x is the variable we need to solve for.
Verify
To check the correctness of a solution by substituting the calculated value back into the original equation or problem statement.
Example:
After solving x + 3 = 7 to get x = 4, you can verify your answer by plugging 4 back in: 4 + 3 = 7, which is true.
Work Problems
Word problems that calculate the time it takes for individuals or machines to complete a task, often working together.
Example:
If one painter can paint a room in 4 hours and another in 6 hours, figuring out how long it takes them working together is a work problem.
Y-intercept
The point where a line crosses the y-axis, representing the initial value or starting point when the independent variable is zero.
Example:
In a graph showing the cost of a taxi ride, the y-intercept would be the initial flat fee before any miles are driven.