This problem tests solving two-step equations from word problems in the form $px + q = r$, where we multiply the variable then add a constant, using inverse operations. The bike rental costs $12 per hour (variable cost) plus a $6 helmet fee (fixed cost), totaling $78, so we write $12h + 6 = 78$ where h is the number of hours. To solve: (1) subtract 6 from both sides to isolate the variable term ($12h = 72$), then (2) divide by 12 to isolate h ($h = 72 ÷ 12 = 6$). The rental lasted 6 hours. Common errors include: stopping after one step (getting 72 as the answer), reversing operations (dividing first: $78 ÷ 12 = 6.5$, then subtracting 6 = $0.5$), or arithmetic mistakes ($72 ÷ 12 = 7$). Strategy: identify the equation form from context (hourly rate × hours + helmet fee = total), apply inverse operations in reverse order (subtract then divide), and verify by substituting back ($12 × 6 + 6 = 72 + 6 = 78 ✓$).

This question tests solving two-step equations from word problems in the px + q = r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px + q = r, first subtract q from both sides to isolate the variable term, for example, 7x + 2 = 44 becomes 7x = 42, then divide both sides by p to isolate the variable, so 7x / 7 = 42 / 7 gives x = 6; always verify by substituting back, like 7 × 6 + 2 = 42 + 2 = 44, which checks out. For this specific problem, use the given equation 7x + 2 = 44, subtract 2 to get 7x = 42, divide by 7 to find x = 6, verify 7 × 6 + 2 = 44, and interpret as 6 notebooks purchased. The correct two-step process yields x = 6, which is choice A. Common errors include one-step only (44 - 2 = 42, forget divide so x = 42), dividing first (44 / 7 ≈ 6.286, subtract 2 ≈ 4.286 nonsense), arithmetic mistake (42 / 7 = 5 or 7 wrongly), sign error (7x = 46 by adding), or misreading context. Strategy: identify the equation form px + q = r from the context, apply inverse operations in reverse order (subtract then divide to undo add then multiply), maintain equality by doing the same to both sides, verify by substituting back to ensure it satisfies the equation and makes sense (6 notebooks at $7 is $42 plus $2 fee totals $44), and check reasonableness (total close to 6 × 7 + small fee). Comparing to arithmetic, work backwards: 44 - 2 = 42, 42 / 7 = 6, same answer; algebra generalizes for any total; avoid mistakes like stopping early or wrong order.

This question tests solving two-step equations from word problems in the px + q = r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px + q = r, first subtract q from both sides to isolate the variable term, for example, 8m + 3 = 59 becomes 8m = 56, then divide both sides by p to isolate the variable, so 8m / 8 = 56 / 8 gives m = 7; always verify by substituting back, like 8 × 7 + 3 = 56 + 3 = 59, which checks out. For this specific problem, set up the equation from the context: 8m + 3 = 59, subtract 3 to get 8m = 56, divide by 8 to find m = 7, verify 8 × 7 + 3 = 59, and interpret as 7 months of subscription. The correct two-step process yields m = 7, which is choice A. Common errors include one-step only (59 - 3 = 56, forgetting to divide so m = 56), operation order reversed (59 / 8 = 7.375, then subtract 3 ≈ 4.375 nonsense), sign error (adding 3 instead: 8m = 62), arithmetic mistake (56 / 8 = 8 wrongly), or setup wrong (3m + 8 = 59 reversing). Strategy: identify the equation form px + q = r from the context, apply inverse operations in reverse order (subtract then divide to undo add then multiply), maintain equality by doing the same to both sides, verify by substituting back to ensure it satisfies the equation and makes sense (7 months at $8 is $56 plus $3 fee totals $59), and check reasonableness (total around 7 × 8 + small fee). Comparing to arithmetic, work backwards: 59 - 3 = 56, 56 / 8 = 7, same answer; algebra generalizes for any total; avoid mistakes like stopping after one step or wrong order.

This question tests solving two-step equations from word problems in the p(x + q) = r form, where you multiply a grouped expression by a coefficient, using inverse operations to isolate the variable. To solve p(x + q) = r, first divide both sides by p to isolate the grouped term, for example, 4(x + 3) = 36 becomes x + 3 = 9, then subtract q to isolate the variable, so x = 9 - 3 = 6; for the px + q = r form, subtract q first then divide by p, and always verify by substituting back, like 4(6 + 3) = 4 × 9 = 36, which checks out in the context of a party bag with 4 times (paid packs plus 3 bonus) totaling $36 value for x paid packs. For this specific problem, the equation is given as 4(x + 3) = 36, divide by 4 to get x + 3 = 9, subtract 3 to find x = 6, verify 4(6 + 3) = 36, and interpret as 6 snack packs paid for. The correct two-step process yields x = 6, which is choice B. Common errors include subtracting first instead of dividing (wrong order: 4x = 33, not integer), dividing only and forgetting to subtract (x + 3 = 9 so x=9 choice C or 12 choice A), sign errors like adding (x = 12), arithmetic mistakes (36 ÷ 4 = 9, then 9 + 3 = 12), or wrong setup though provided. Strategy tips include identifying the p(x + q) = r form from multiplied group with added bonus items, applying inverse operations in reverse (divide then subtract), maintaining equality by doing the same to both sides, verifying by substitution and context sense (6 paid plus 3 bonus is 9 at $4 each totals $36, reasonable), and checking reasonableness. Compared to arithmetic working backwards (36 ÷ 4 = 9, 9 - 3 = 6), algebra generalizes for any total; avoid mistakes like stopping after one step, wrong order, sign errors, or mismatched setup.

This question tests solving two-step equations from word problems in the px + q = r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px + q = r, first subtract q from both sides to isolate the variable term, for example, 12t + 3 = 75 becomes 12t = 72, then divide both sides by p to isolate the variable, so 12t ÷ 12 = 72 ÷ 12 gives t = 6; for the other form p(x + q) = r, divide by p first then subtract q, and always verify by substituting back, like 12 × 6 + 3 = 72 + 3 = 75, which checks out in the context of museum tickets at $12 each plus $3 fee totaling $75 for t tickets. For this specific problem, the equation is given as 12t + 3 = 75, subtract 3 to get 12t = 72, divide by 12 to find t = 6, verify 12 × 6 + 3 = 75, and interpret as 6 tickets bought. The correct two-step process yields t = 6, which is choice A. Common errors include subtracting only and forgetting to divide (t = 72, choice D), reversing order by dividing first (75 ÷ 12 = 6.25 then subtract 3, choice C), sign errors like adding instead (12t = 78), arithmetic mistakes (72 ÷ 12 = 5), or wrong setup though it's provided. Strategy tips include identifying the px + q = r form from per-ticket cost plus fixed fee, applying inverse operations in reverse (subtract then divide), maintaining equality by doing the same to both sides, verifying by substitution and context sense (6 tickets at $12 is $72 plus $3 totals $75, reasonable), and checking reasonableness. Compared to arithmetic working backwards (75 - 3 = 72, 72 ÷ 12 = 6), algebra generalizes for any total; avoid mistakes like stopping after one step, wrong order, sign errors, or mismatched setup.

This question tests solving two-step equations from word problems in the px + q = r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px + q = r, first subtract q from both sides to isolate the variable term, for example, 8.75x + 10 = 80 becomes 8.75x = 70, then divide both sides by p to isolate the variable, so 8.75x ÷ 8.75 = 70 ÷ 8.75 gives x = 8; for the other form p(x + q) = r, divide by p first then subtract q, and always verify by substituting back, like 8.75 × 8 + 10 = 70 + 10 = 80, which checks out in the context of cookie dough tubs at $8.75 each plus $10 fee totaling $80 for x tubs. For this specific problem, the equation is given as 8.75x + 10 = 80, subtract 10 to get 8.75x = 70, divide by 8.75 to find x = 8, verify 8.75 × 8 + 10 = 80, and interpret as 8 tubs bought. The correct two-step process yields x = 8, which is choice A. Common errors include subtracting only and forgetting to divide (x = 70, choice C), reversing order by dividing first (80 ÷ 8.75 ≈ 9.14 then subtract 10, nonsensical), sign errors like adding instead (8.75x = 90), arithmetic mistakes (70 ÷ 8.75 = 9 or 6), or wrong setup though it's provided. Strategy tips include identifying the px + q = r form from per-tub cost plus fixed fee, applying inverse operations in reverse (subtract then divide), maintaining equality by doing the same to both sides, verifying by substitution and context sense (8 tubs at $8.75 is $70 plus $10 totals $80, reasonable), and checking reasonableness. Compared to arithmetic working backwards (80 - 10 = 70, 70 ÷ 8.75 = 8), algebra generalizes for any total; avoid mistakes like stopping after one step, wrong order, sign errors, or mismatched setup.

This question tests verifying solutions to two-step equations from word problems in the px + q = r form, solving with inverse operations and checking fit. To solve 18h + 6 = 78, subtract 6 from both sides: 18h = 72, then divide by 18: h = 4; verify by substituting: 18 × 4 + 6 = 72 + 6 = 78, which checks out in the context of $18 per hour plus $6 fee totaling $78. For this specific problem, the equation is given as 18h + 6 = 78, solve step 1 by subtracting 6 to get 18h = 72, step 2 by dividing by 18 to get h = 4, verify 18 × 4 + 6 = 78, and interpret as 4 hours tutored. The correct value is h = 4, which is choice B. Common errors include solutions like h = 3 giving 18 × 3 + 6 = 60 ≠ 78, or h = 13/3 ≈ 4.33 giving 18 × (13/3) + 6 = 78 + 6 = 84 ≠ 78, or h = 7/2 = 3.5 giving 18 × 3.5 + 6 = 69 ≠ 78, or arithmetic mistakes in verification. Strategy tips include identifying the form px + q = r, applying inverse operations by subtracting then dividing, maintaining equality, verifying by substituting back to ensure equality and contextual sense (4 hours at $18 is $72 plus $6 = $78), and checking reasonableness for positive integer hours. Comparing to arithmetic, working backwards by subtracting 6 then dividing by 18 matches; avoid mistakes like partial solving or ignoring the verification step.

This problem tests solving two-step equations from word problems in the form px+q=r, where we multiply the variable then add a constant, using inverse operations. The club charges $2.25 per week (variable cost) plus a $4.50 sign-up fee (fixed cost), totaling $24.75, so we write 2.25w + 4.50 = 24.75 where w is the number of weeks. To solve: (1) subtract 4.50 from both sides to isolate the variable term (2.25w = 20.25), then (2) divide by 2.25 to isolate w (w = 20.25 ÷ 2.25 = 9). The student paid for 9 weeks. Common errors include: stopping after one step (getting 20.25 as the answer), reversing operations (dividing first: 24.75÷2.25=11, then subtracting 4.50=6.50), or arithmetic mistakes (20.25÷2.25=8). Strategy: identify the equation form from context (weekly rate × weeks + fixed fee = total), apply inverse operations in reverse order (subtract then divide), and verify by substituting back (2.25×9+4.50=20.25+4.50=24.75✓).

This problem tests solving two-step equations from word problems in the form $px + q = r$, where we multiply the variable then add a constant, using inverse operations. The store charges $8 per game (variable cost) plus a $5 membership fee (fixed cost), totaling $61, so we write $8g + 5 = 61$ where g is the number of games. To solve: (1) subtract 5 from both sides to isolate the variable term ($8g = 56$), then (2) divide by 8 to isolate g ($g = 56 ÷ 8 = 7$). You bought 7 games. Common errors include: stopping after one step (getting 56 as the answer), reversing operations (dividing first: $61 ÷ 8 = 7.625$, then subtracting 5=2.625), or arithmetic mistakes ($56 ÷ 8 = 6$ or 8). Strategy: identify the equation form from context (price per game × games + membership fee = total), apply inverse operations in reverse order (subtract then divide), and verify by substituting back ($8 × 7 + 5 = 56 + 5 = 61 ✓$).

This problem tests solving two-step equations from word problems in the form $px + q = r$, where we multiply the variable then add a constant, using inverse operations. The streaming service costs $9.75 per month (variable cost) plus a $4.50 fee (fixed cost), totaling $53.25, so we write $9.75m + 4.50 = 53.25$ where m is the number of months. To solve algebraically: (1) subtract 4.50 from both sides ($9.75m = 48.75$), then (2) divide by 9.75 ($m = 48.75 ÷ 9.75 = 5$). Working backward gives the same result: $$53.25 - $4.50 = \$48.75$, then $\$48.75 ÷ $9.75 = 5$ months. Common errors include: stopping after one step, reversing operations, or arithmetic mistakes with decimals. Both methods—algebraic equation solving and working backward—yield 5 months, demonstrating that algebra generalizes the arithmetic approach of undoing operations in reverse order.