The population of Town A after y years will be its initial population plus the growth: 12000 + 250y. The population of Town B after y years will be its initial population minus the decrease: 15000 - 150y. To find when the populations are equal, we set these two expressions equal to each other: 12000 + 250y = 15000 - 150y.

Let the width be w. The length is twice the width, so L = 2w. The formula for the perimeter of a rectangle is P = 2L + 2w. Substituting L = 2w and P = 120 into the formula gives 120 = 2(2w) + 2w. This can also be written as 2w + 2(2w) = 120.

Let the angle be x. Its supplement is 180 - x. 'One-third the measure of its supplement' is (1/3)(180 - x). '18 degrees more than' this quantity is (1/3)(180 - x) + 18. Since this is equal to the measure of the angle x, the equation is x = (1/3)(180 - x) + 18.

The final temperature (13°F) is equal to the initial temperature (-8°F) plus the total increase. The total increase is the rate (3°F per hour) multiplied by the number of hours (h), which is 3h. So, the equation is -8 + 3h = 13.

This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes a train traveling at 80 miles per hour for h hours, requiring an equation that shows distance equals rate times time (D = rt). Choice C is correct because D = 80h accurately models the relationship where distance equals the rate (80 mph) multiplied by time (h hours). Choice A incorrectly adds instead of multiplying, Choice B reverses the relationship making hours equal 80 times distance, and Choice D uses an incorrect rate of 8 instead of 80. To help students, reinforce the fundamental formula distance = rate × time and practice identifying which quantity is which in word problems. Common errors include confusing addition with multiplication in rate problems.

This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes a store with $150 fixed costs that makes $9 profit per product, requiring an equation of the form R = fixed costs + (profit per product × products). Choice B is correct because R = 150 + 9p accurately models starting with $150 in fixed costs and adding $9 profit for each of p products sold. Choice A incorrectly reverses the coefficients making it 150p + 9, Choice C subtracts the profit instead of adding it, and Choice D omits the fixed costs entirely. To help students, teach them to identify fixed versus variable components and understand that profits typically add to totals. Business contexts like this help students see real-world applications of linear equations.

This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes starting with $120 and adding $25 each month for m months, requiring an equation of the form T = initial amount + (monthly amount × months). Choice A is correct because T = 120 + 25m accurately models starting with $120 and adding $25 for each of m months. Choice B incorrectly reverses the constants, Choice C subtracts instead of adds, and Choice D doesn't include the variable m. To help students, emphasize identifying the starting value and the repeated action, then translating 'each month' or 'per month' into multiplication. Regular practice with savings and accumulation problems builds this skill.

This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes using 8 grams of Chemical A (fixed amount) plus 3 grams per trial of Chemical B, requiring an equation of the form G = fixed amount + (amount per trial × trials). Choice A is correct because G = 8 + 3t accurately models using 8 grams initially plus 3 grams for each of t trials. Choice B incorrectly groups the constants before multiplying, Choice C subtracts instead of adds, and Choice D reverses the coefficients. To help students, teach them to distinguish between fixed amounts (used once) and variable amounts (used repeatedly). Practice identifying keywords like 'per trial' or 'each time' that signal multiplication.

This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes a constant speed of 18 miles per hour multiplied by time t to find distance d, requiring an equation of the form d = speed × time. Choice B is correct because it accurately models the relationship using multiplication of the given speed and time variable. Choice C is incorrect because it divides instead of multiplies, a common error when students invert the rate relationship. To help students, teach identifying key parts of scenarios like velocity and time, and translating them into mathematical terms. Encourage practice with varied contexts such as cycling or driving to build flexibility in model creation.

This question tests middle-level math skills: choosing an equation to model a situation. Understanding equations involves recognizing relationships between variables and constants as described in a scenario. In this example, the scenario describes a constant speed of 55 miles per hour multiplied by time t to find distance d, requiring an equation of the form d = speed × time. Choice B is correct because it accurately models the relationship using multiplication of the given speed and time variable. Choice A is incorrect because it uses division instead of multiplication, a common error when students confuse speed with rate formulas. To help students, teach identifying key parts of scenarios like rates and variables, and translating them into mathematical terms. Encourage practice with varied contexts such as travel or work rates to build flexibility in model creation.