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Pre-Algebra
Table of Contents
Unit 9 Overview: Math Models and Geometry
9.1 Use a Problem Solving Strategy
9.2 Solve Money Applications
9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
9.4 Use Properties of Rectangles, Triangles, and Trapezoids
9.5 Solve Geometry Applications: Circles and Irregular Figures
9.6 Solve Geometry Applications: Volume and Surface Area
9.7 Solve a Formula for a Specific Variable

➕pre-algebra review

9.7 Solve a Formula for a Specific Variable

Citation:

Formulas and variables are the building blocks of algebra. They help us understand relationships between quantities and solve real-world problems. From calculating travel times to figuring out interest rates, these tools are incredibly useful.

By learning how to work with formulas and variables, you'll gain the power to analyze and predict outcomes in various situations. Whether it's rearranging equations or interpreting what each variable means, these skills will serve you well in math and beyond.

Formulas and Variables

Distance, rate and time applications

  • The distance, rate, and time formula $d = rt$ describes the relationship between these three variables
    • $d$ represents the total distance traveled (miles, kilometers)
    • $r$ represents the constant rate or speed of travel (mph, kph)
    • $t$ represents the time spent traveling at the constant rate (hours, minutes)
  • Find the distance by multiplying the rate and time
    • A car traveling at 60 mph for 2 hours covers a distance of $d = 60 \times 2 = 120$ miles
    • A person walking at 3 kph for 4 hours travels a distance of $d = 3 \times 4 = 12$ km
  • Calculate the rate by dividing the distance by the time
    • If a person walks 6 miles in 2 hours, their rate is $r = \frac{6}{2} = 3$ mph
    • A train that travels 450 km in 3 hours has a rate of $r = \frac{450}{3} = 150$ kph
  • Determine the time by dividing the distance by the rate
    • A train traveling 300 miles at 50 mph takes $t = \frac{300}{50} = 6$ hours
    • An airplane flying 2,400 km at 800 kph takes $t = \frac{2400}{800} = 3$ hours

Rearranging formulas for variables

  • Isolate a variable by applying the same operation to both sides of the equation
    • Addition or subtraction moves terms to the opposite side with the sign changed
      • Rearrange $a + b = c$ to $a = c - b$ by subtracting $b$ from both sides
      • Rearrange $x - 3 = 7$ to $x = 7 + 3$ by adding 3 to both sides
    • Multiplication or division cancels out factors on the opposite side
      • Rearrange $xy = z$ to $x = \frac{z}{y}$ by dividing both sides by $y$
      • Rearrange $\frac{a}{4} = b$ to $a = 4b$ by multiplying both sides by 4
  • Divide both sides by the coefficient to isolate a variable multiplied by a number
    • Rearrange $2x = 10$ to $x = \frac{10}{2} = 5$ by dividing both sides by 2
    • Rearrange $7y = 21$ to $y = \frac{21}{7} = 3$ by dividing both sides by 7
  • Multiply both sides by the denominator to isolate a variable in a fraction
    • Rearrange $\frac{a}{b} = c$ to $a = bc$ by multiplying both sides by $b$
    • Rearrange $\frac{x}{5} = 3$ to $x = 3 \times 5 = 15$ by multiplying both sides by 5
  • Use substitution to replace a variable with its equivalent expression when solving complex formulas

Contextual meaning of formula variables

  • Variables represent unknown or changing quantities within the context of a formula (algebra)
    • In the area formula $A = lw$, $A$ is the area, $l$ is the length, and $w$ is the width
    • For the volume of a cylinder $V = \pi r^2 h$, $V$ is volume, $r$ is radius, $h$ is height
  • Physics formulas use variables to represent measurable quantities
    • $d$ often represents distance or displacement (meters, feet)
    • $v$ represents velocity or speed (m/s, ft/s)
    • $a$ represents acceleration (m/s², ft/s²)
    • $t$ represents time (seconds, minutes)
  • Geometry formulas use variables for dimensions and measurements
    • $A$ often represents the area of a shape (m², ft²)
    • $V$ represents the volume of a 3D object (m³, ft³)
    • $r$ or $R$ represents the radius of a circle or sphere (cm, in)
    • $h$ represents the height of a figure (cm, in)
  • Interpret the meaning of each variable based on the problem's context and the formula's application
    • A formula for the cost of a rental car $C = rt + f$ has $C$ for total cost, $r$ for hourly rate, $t$ for rental time, $f$ for fixed fees
    • The compound interest formula $A = P(1 + \frac{r}{n})^{nt}$ uses $A$ for final amount, $P$ for principal, $r$ for annual interest rate, $n$ for compounding frequency, $t$ for time in years

Understanding Algebraic Equations and Formulas

  • An equation is a mathematical statement that shows two expressions are equal
  • A formula is a special type of equation that expresses a relationship between variables
  • To solve an equation or formula, manipulate it to isolate the desired variable on one side

Key Terms to Review (21)

Commutative Property: The commutative property is a fundamental mathematical principle that states the order in which two numbers are combined does not affect the final result. This property applies to addition and multiplication operations, allowing the terms to be rearranged without changing the outcome.
Constant: A constant is a quantity in an algebraic expression or equation that has a fixed, unchanging value. It is a value that does not vary or change throughout the given context or problem.
Equation: An equation is a mathematical statement that expresses the equality between two expressions, typically involving variables and constants. Equations are fundamental in algebra and are used to represent and solve various problems by finding the unknown values.
Variable: A variable is a symbol, typically a letter, that represents an unknown or changeable quantity in an algebraic expression or equation. It is a fundamental concept in algebra that allows for the generalization of mathematical relationships and the solution of problems involving unknown values.
Linear Equation: A linear equation is a mathematical equation in which the variables are raised only to the first power and are connected by addition, subtraction, or equality. These equations represent a straight line when graphed on a coordinate plane.
Inverse Operations: Inverse operations are mathematical operations that undo or reverse the effects of another operation. They are used to solve equations, simplify expressions, and perform calculations in various mathematical contexts.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the scale or magnitude of the variable, indicating how much of that variable is present in the expression.
Distributive Property: The distributive property is a fundamental algebraic rule that states the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions by distributing a factor across multiple terms within a parenthesis or other grouping symbol.
Cross Multiplication: Cross multiplication is a technique used to solve proportions and equations involving fractions. It involves multiplying the numerator of one fraction with the denominator of the other fraction, and vice versa, to find a missing value in the proportion or equation.
Substitution: Substitution is the process of replacing one value or expression with another in an equation or formula. It is a fundamental technique used to solve equations, simplify expressions, and manipulate formulas to isolate a specific variable.
Rate: A rate is a measure of change or a comparison between two related quantities, often expressed as a ratio or fraction. It describes the speed or pace at which something occurs or changes over time.
Factoring: Factoring is the process of breaking down a polynomial expression into a product of smaller, simpler expressions. It involves identifying common factors among the terms and expressing the polynomial as a product of those factors.
Distance Formula: The distance formula is a mathematical equation used to calculate the straight-line distance between two points in a coordinate plane. It is a fundamental concept in geometry and algebra that allows for the precise measurement of the length between any two points on a coordinate grid.
Dependent Variable: The dependent variable is the variable that is being measured or observed in an experiment or study. It is the outcome or result that is influenced or affected by the independent variable.
Algebra: Algebra is a branch of mathematics that deals with the study of mathematical expressions and the relationships between them. It involves the use of variables, equations, and functions to represent and solve problems. Algebra is a fundamental tool in pre-algebra and other areas of mathematics, as it provides a systematic way to work with unknown quantities and to generalize patterns and relationships.
Isolate: Isolate, in the context of solving a formula for a specific variable, refers to the process of manipulating an equation to separate or extract the desired variable from the other terms, allowing for its explicit solution.
Literal Equation: A literal equation is a mathematical equation in which the variables are represented by letters or symbols, rather than numerical values. These equations express relationships between variables and are commonly used to solve for a specific variable when the values of the other variables are known.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers. The formula provides a systematic way to find the roots or solutions of a quadratic equation.
Solve: To solve a problem or equation means to find the value or values of the unknown variable(s) that satisfy the given conditions or make the equation true. Solving is a fundamental mathematical skill that involves using various strategies and techniques to determine the solution to a problem.
Formula: A formula is a mathematical expression that represents a relationship between variables or quantities. It is a concise way to express a rule or principle that can be used to solve problems or make calculations.
Independent Variable: The independent variable is the variable in a study or experiment that the researcher manipulates or changes to observe its effect on the dependent variable. It is the variable that the researcher believes will influence or cause changes in the dependent variable.