Algebra 1 Unit 2: Linear Equations, Constraints, and Equivalence

Comprehensive vocabulary flashcards covering linear modeling, constraints, inequalities, equivalent equations, and formula manipulations from Unit 2.

Total Pizza Cost Expression

The algebraic expression $6c + 10f$, which represents the total cost in dollars of ordering $c$ cheese pizzas at $\$6$ each and $f$ four-topping pizzas at $\$10$ each.

Pizza Cost Equation (Two Variables)

The equation $5c + 6d = T$, representing the total cost $T$ of ordering $c$ large cheese pizzas at $\$5$ each and $d$ large one-topping pizzas at $\$6$ each.

Constraint Equations (Milk and Cookies)

Equations representing quantities and limits for $15$ people receiving $12\,\text{ounces}$ of milk and $4$ cookies each, including $m = 12(15)$, $4n = c$, and budget equation $b = 2(3) + 3(4.50)$.

Split-Pace Distance Equation

An equation that calculates total distance $D$ in miles from speeds and times in minutes converted to hours, such as $D = 8\left(\frac{a}{60}\right) + 7.5\left(\frac{b}{60}\right)$ or $D = M\left(\frac{5}{60}\right) + N\left(\frac{15}{60}\right)$.

Total Trophy Cost Expression

The algebraic expression $5.99g + 6.49u$, representing the total dollar cost of ordering $g$ gold baseball trophies at $\$5.99$ each and $u$ uniform baseball trophies at $\$6.49$ each.

Equal Contribution Expression

The algebraic expression $\frac{350}{x}$, representing the amount in dollars that each of $x$ students must fundraise to contribute equally toward a $\$350$ equipment purchase.

Team Score Inequality (Trivia Contest)

The compound inequality $0 \le T \le 50$, which describes the total score $T$ achievable in one round by a $5$-member team where each member can score up to $10$ points.

Constrained Score Range

The compound inequality $30 \le T \le 40$, representing the total points earned by a $5$-player team when each player scores between $6$ and $8$ points.

Total Weight Equation (Crushed Stone)

The proportional equation $W = 2{,}000n$, representing the total weight in pounds $W$ of $n$ loads of crushed stone when each load weighs $2{,}000\,\text{pounds}$.

Equal Distribution Formula (Band Fundraiser)

The equation $A = \frac{2{,}000}{m}$, which represents the dollar amount $A$ each member receives when a $\$2{,}000$ fundraising total is divided equally among $m$ members.

Teaspoon-to-Cup Relationship

The equation $t = 48C$, which calculates the number of teaspoons $t$ in $C$ cups based on $1\,\text{cup} = 16\,\text{tablespoons}$ and $1\,\text{tablespoon} = 3\,\text{teaspoons}$.

Bushel-to-Quart Relationship

The equation $32b = q$, representing the relationship between the number of bushels $b$ and the number of quarts $q$, given that $1\,\text{bushel} = 4\,\text{pecks}$ and $1\,\text{peck} = 8\,\text{quarts}$.

Linear Account Balance Equation

The equation $225 - 20w = d$, representing the remaining balance $d$ in dollars after taking out $\$20$ per week for $w$ weeks from an initial $\$225$.

Constraint Equations (Poetry Club Meeting)

The equations modeling servings and budget for $12$ members, including $f = 8 \cdot 12$, $2p = c$, and $2(3.99) + 2.50 = b$, based on 8 oz punch and 2 oz cheese per person.

Density Formula

The relationship $D = \frac{m}{v}$, which defines density $D$ in grams per cubic centimeter ($\text{g/cm}^3$) as mass $m$ in grams divided by volume $v$ in cubic centimeters.

Solution to a Two-Variable Equation

An ordered pair of numerical values that makes the equation true, such as $(2, 5)$ and $(4, 2)$ for $2.25n + 1.50b = 12$, or $(0, 445)$ and $(118, 209)$ for $0.10d + 0.05n = 22.25$.

Transportation Constraint Equation

The equation $4c + 6v = 80$, representing the combination of $c$ cars (seating $4$ students) and $v$ vans (seating $6$ students) needed to seat exactly $80$ students.

Setup Fee

The constant initial fee in a cost function that is independent of quantity ordered, such as the $\$40$ fee in $21x + 40 = 187$ (where each shirt costs $x = \$7$) or the $\$65$ fee in $201.50 = f + 6.50(21)$.

Discrete Gathering Budget Equation

The budget calculation $b = 1.50(6) + 2.99(2)$, representing the total cost for $10$ people who need $6$ whole bottles of sparkling water (120 oz total) at $\$1.50$ and $2$ whole packages of pretzels (30 oz total) at $\$2.99$.

Speed Formula

The relationship $S = \frac{d}{t}$, representing speed $S$ as distance traveled divided by time taken, such as $S = \frac{100}{2.5}$ for $100\,\text{feet}$ in $2.5\,\text{seconds}$.

Per-Student Distribution Expression

The expression $\frac{12 \cdot 12}{x}$, which represents the number of donuts each student receives when $12$ dozen donuts are distributed equally among $x$ students.

Graph of a Linear Equation

The collection of all coordinate pairs $(x, y)$ that satisfy the equation, such as points $(-1, 1.5)$ and $(0, 3)$ satisfying $4y - 6x = 12$, or $(0, 2)$ and $(3, 1)$ satisfying $x + 3y = 6$.

Meaning of a Coordinate Point in Context

The practical interpretation of a point $(x, y)$ on a line; for example, $(30, 7)$ on the ticket line $8x + 5y = 275$ represents selling $30$ adult tickets and $7$ children's tickets.

Savings Accumulation Equation

The linear relationship $y = 50 + 20x$, describing a bank balance $y$ after starting with $\$50$ and depositing $\$20$ per week for $x$ weeks, reaching $\$250$ at week $10$ ($(10, 250)$).

Team Size Constraint Inequality

The inequality $2 \le n \le 4$, which defines the allowable number of players $n$ permitted on a team in a video game.

Game Player Range Inequality

The compound inequality $16 \le n \le 32$, representing the possible total number of players in a video game with $8$ teams having between $2$ and $4$ players each.

Running Distance Equation with Minute-to-Hour Conversion

The equation $d = 5.4\left(\frac{m}{60}\right) + b\left(\frac{n}{60}\right)$, describing total distance $d$ in miles when running at $5.4\,\text{mph}$ for $m$ minutes and $b\,\text{mph}$ for $n$ minutes.

Historical Length Conversion Equations

The mathematical relationships $c = 4.5h$ (or $1\,\text{cubit} = 4.5\,\text{hands}$) and $p = 7.5h$ (or $1\,\text{pace} = 7.5\,\text{hands}$), derived from $9\,\text{hands} = 2\,\text{cubits}$ and $5\,\text{cubits} = 3\,\text{paces}$.

Equivalent Savings Models

Alternative equations modeling a savings balance that grows by $\$100$ each month starting from $\$700$ at month $0$ ($\$1{,}200$ at month $5$), including $A = 700 + 100m$, $A - 700 = 100m$, and $A = 1{,}200 + 100(m - 5)$.

Equivalent Equations

Equations that share the same solution set, such as $6x + 9 = 12$ and $2x + 3 = 4$ (obtained by dividing by $3$), or $0.05n + 0.1d = 3.65$ and $5n + 10d = 365$ (obtained by multiplying by $100$).

Same Solution Property

The property that reversible arithmetic operations preserve the truth of an equation; for example, $2x - 5 = 15$ has the exact same solution ($x = 10$) as $2x = 20$, $2x - 20 = 0$, and $4x - 10 = 30$.

Banana Unit Price Equation

The direct variation equation $P = 0.59n$, representing the total price $P$ in dollars for $n$ bananas at a constant rate of $\$0.59$ per banana.

Coin Jar Constraint Equation

The equation $0.1d + 0.25q = 10$, representing combinations of $d$ dimes and $q$ quarters totaling $\$10.00$, with solutions including $(100, 0)$, $(50, 20)$, and $(10, 36)$.

Graph Solution Point in Context (Jam Ingredients)

A coordinate pair on the graph of $1.80s + 2.50b = 19.40$ (identified as Point C) that represents a combination of pounds of sugar $s$ and pounds of strawberries $b$ that exactly costs $\$19.40$.

Distributive Property in Equation Solving

The algebraic property $a(b + c) = ab + ac$ used to expand or factor expressions, such as transforming $3(x + 6) = 4x + 7$ into $3x + 18 = 4x + 7$, or rewriting $5x + 30 = 45$ as $5(x + 6) = 45$.

Multiplicative Equivalence of Equations

The principle that multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation, such as dividing $5(x + 4) = 100$ by $5$ to yield $x + 4 = 20$.

Delivery Fee in a Linear Model

The fixed constant term $x$ in the cost model $320 = x + 7.50(36)$, representing the delivery fee charged by the second store when ordering $36$ potted plants at $\$7.50$ each.

Shirt Cost Formula (Environmental Science Club)

The linear equation $162.50 = 15x + 20$, solved by isolating $x$ to determine that the printing company charges $x = \$9.50$ per shirt for $15$ members after a $\$20$ setup fee.

Celsius-to-Fahrenheit Conversion Formula

The formula $C = \frac{5}{9}(F - 32)$, which calculates temperature in degrees Celsius $C$ from degrees Fahrenheit $F$, yielding approximately $15.6^\circ\text{C}$ when $F = 60^\circ\text{F}$.

Age Relationship Equivalence

The algebraic equivalence between $d = 2s + 5$ and $d - 5 = 2s$, representing that Diego's age $d$ is $5$ years more than twice his sister's age $s$.

Rearranged Trail Mix Equation

The isolated equation $r = \frac{11.70 - 5.20a}{2.75}$, derived from $5.20a + 2.75r = 11.70$, used to directly compute the pounds of raisins $r$ based on pounds of almonds $a$.

Explicit Form of a Linear Equation

The solved form of a two-variable equation for a single variable, transforming $2x + 4y - 31 = 123$ into $x = 77 - 2y$ when solving for $x$, or $y = 38.5 - 0.5x$ when solving for $y$.

Meat Budget Rearrangements

Equivalent forms of the budget equation $0.9c + 1.89r = 17.01$ solved for each variable: $c = 18.9 - 2.1r$ (to find pounds of chicken) and $r = -\frac{10}{21}c + 9$ (to find pounds of ribs).

Constant Distance Rate Formula

The formula $r = \frac{180}{t}$, which determines the travel speed $r$ in miles per hour needed to complete a constant $180\,\text{mile}$ trip in $t$ hours.

Fruit Purchase Constraint Solutions

Coordinate pairs $(b, a)$ satisfying $0.50b + 1.00a = 3.50$ for bananas at $\$0.50$ and apples at $\$1.00$, including $(3, 2)$, $(1, 3)$, and $(5, 1)$.