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Difficulty: MediumLinear Equations in Two Variables

A chemist cools a liquid compound in a laboratory refrigerator. The temperature of the liquid decreases at a constant rate. After 44 minutes of cooling, the temperature of the liquid is 22∘C22^\circ\text{C}. After 1010 minutes of cooling, the temperature of the liquid is 13∘C13^\circ\text{C}. Which of the following equations represents the relationship between the temperature, TT, in degrees Celsius, of the liquid and the number of minutes, mm, it has been cooling?

  1. A
    2m+3T=742m + 3T = 74
  2. B
    T=4m+22T = 4m + 22
  3. 3m+2T=563m + 2T = 56Answer
  4. D
    2T−3m=562T - 3m = 56

Answer

The equation 3m+2T=563m + 2T = 56 correctly represents the relationship between the temperature TT and the cooling time mm.
The relationship between the temperature TT and the cooling time mm is linear, which can be represented by the equation T=−1.5m+28T = -1.5m + 28. Rearranging this equation by adding 1.5m1.5m to both sides yields 1.5m+T=281.5m + T = 28. Multiplying the entire equation by 22 to clear the decimal gives 3m+2T=563m + 2T = 56, which is the correct linear relationship.

Step-by-Step Solution

1
Calculate the rate of temperature change (slope, kk) per minute of cooling.
k=13−2210−4=−96=−1.5∘Ck = \frac{13 - 22}{10 - 4} = \frac{-9}{6} = -1.5^\circ\text{C} per minute.
A linear relationship has a constant rate of change, which is found by dividing the change in temperature by the change in time.
2
Use the point-slope form to write the equation of the line.
T−22=−1.5(m−4)T - 22 = -1.5(m - 4), which simplifies to T=−1.5m+28T = -1.5m + 28.
This models the relationship between TT and mm by using one of the given data points and the calculated slope.
3
Rearrange the equation into standard form with integer coefficients.
1.5m+T=28  ⟹  3m+2T=561.5m + T = 28 \implies 3m + 2T = 56.
The options are written in standard form, so adding 1.5m1.5m to both sides and multiplying the entire equation by 22 matches the target format.

Key Concept

Linear Equations in Two Variables
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Topics

Linear Equations in Two Variables

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