This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "750 ml divided equally into p cups" → expression 750/p (p any positive whole number representing cups, 750/p gives amount per cup for any p). The correct choice is C, $\dfrac{750}{p}$, because p represents any positive whole number of cups, and the expression correctly divides the total water by p to find the general amount per cup. A common error is choosing A, $750p$, which multiplies instead of divides, reversing the operation, or B, $750-p$, which subtracts, misinterpreting division as subtraction in sharing equally. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "rectangle with length l and width w" → expression 2l+2w (l and w any positive numbers representing dimensions, 2l+2w gives perimeter for any l and w). The correct choice is B, $2l+2w$, where l and w are general variables for any dimensions, and the expression models the perimeter by adding twice each side. A common error is choosing A, $lw$, which is area instead of perimeter, or D, $2lw$, which doubles the area. Defining variables: state what variable represents (let l=length in cm, let w=width in cm—clear definition prevents confusion). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values).

This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "24 ft rope cut into 6 equal pieces, p length per piece" → equation 6p = 24 (p is unknown specific length, solve: p=4). The correct choice is C, $6p=24$, because p represents a specific unknown length, and the equation correctly multiplies pieces by p to equal total, allowing solution for p. A common error is choosing B, $\dfrac{p}{6}=24$, which divides incorrectly, or A, $p-6=24$, which subtracts, both using wrong operations for equal division. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "temperature t between 18°C and 30°C" → variable t represents any number in the set [18,30] (general range). The correct choice is B, $t$ can be any number such that $18\le t\le 30$, because t represents any value in that closed interval, correctly describing the general set of possible temperatures without implying specifics or negatives. A common error is choosing A, $t$ can be any whole number greater than 30, which exceeds the upper limit, or C, $t$ must be a negative number, which ignores the positive range, both treating the variable's purpose as unrestricted or wrong in set definition. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "rectangle with length l and width w" → expression 2l + 2w (l and w any positive numbers representing dimensions, 2l + 2w gives perimeter for any l and w). The correct choice is B, $2l+2w$, because l and w represent any positive lengths, and the expression correctly adds twice the length and twice the width to give the general perimeter formula. A common error is choosing A, $lw$, which is the area instead of perimeter, confusing the two concepts, or D, $2lw$, which doubles the area formula, misapplying the operations for perimeter. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "bus travels 40 mph for t hours" → equation d = 40t (t any non-negative number representing time, d = 40t gives distance for any t). The correct choice is A, $d=40t$, because t represents any time, and the equation correctly multiplies speed by time to express the general distance formula. A common error is choosing D, $d=\dfrac{40}{t}$, which divides instead, confusing distance with another rate, or B, $t=40d$, which solves for t incorrectly, swapping variables. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "$10 per pizza, $80 to spend, p pizzas" → equation 10p=80 (p is unknown specific number, solve: p=8). The correct choice is B, $10p=80$, where p is a specific unknown, and the equation multiplies cost per pizza by number to equal budget. A common error is choosing A, $10+p=80$, which adds instead of multiplying, or D, $80p=10$, which reverses. Defining variables: state what variable represents (let p=number of pizzas—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: p in 10p=80 has one answer p=8, specific). Writing from context: read problem (identifies relationship: cost times number equals total), write equation (10p=80).

This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the examples are Situation 1: n any non-negative whole number for notebooks (general set), Situation 2: x unknown specific starting money (solve for specific value). The correct choice is B, because it accurately distinguishes n as general for any notebooks and x as specific unknown to solve for, matching the purposes in each context. A common error is choosing A, treating both as specific unknowns, or C, swapping the purposes, both confusing general variables with specific ones. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "$12 sign-up plus $5 per visit v" → expression 12 + 5v (v any non-negative whole number representing visits, 12 + 5v gives total cost for any v). The correct choice is C, $12+5v$, because v represents any number of visits, and the expression correctly adds the fixed fee to the variable cost per visit for a general total. A common error is choosing A, $5(v+12)$, which distributes incorrectly, or B, $12v+5$, which swaps the coefficients, both misapplying the fixed and variable parts of the cost. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "bus travels 40 mph for t hours" → equation d = 40t (t any non-negative number representing time, d = 40t gives distance for any t). The correct choice is A, $d=40t$, because t represents any time, and the equation correctly multiplies speed by time to express the general distance formula. A common error is choosing D, $d=\dfrac{40}{t}$, which divides instead, confusing distance with another rate, or B, $t=40d$, which solves for t incorrectly, swapping variables. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).