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Difficulty: MediumSystems of Linear Equations

A scientist is monitoring the population of two species of bacteria, Species A and Species B, in a controlled environment. Initially, the combined population of the two species is 12,00012,000. Over the next week, the population of Species A doubles, and the population of Species B triples. If the total combined population of both species is 29,00029,000 at the end of the week, what was the initial population of Species A?

Answer: 7000

Answer

The initial population of Species A was 7000.
To find the initial population of Species A, we set up a system of linear equations representing the total initial population and the population after one week. Let xx represent the initial population of Species A and yy represent the initial population of Species B. The initial combined population is given by x+y=12,000x + y = 12,000. After one week, the population of Species A doubles (2x2x) and Species B triples (3y3y), so the new combined population is 2x+3y=29,0002x + 3y = 29,000. We can express yy in terms of xx as y=12,000−xy = 12,000 - x. Substituting this expression into the second equation gives 2x+3(12,000−x)=29,0002x + 3(12,000 - x) = 29,000. Distributing and simplifying yields −x+36,000=29,000-x + 36,000 = 29,000, which simplifies to x=7,000x = 7,000. Therefore, the initial population of Species A was 7,000.

Step-by-Step Solution

1
Define the variables and set up the first equation.
x+y=12,000x + y = 12,000
Let xx be the initial population of Species A and yy be the initial population of Species B. Their combined initial population is 12,00012,000.
2
Set up the second equation based on the growth after one week.
2x+3y=29,0002x + 3y = 29,000
The population of Species A doubles to 2x2x and Species B triples to 3y3y, summing to a total of 29,00029,000.
3
Solve for xx using substitution.
x=7,000x = 7,000
From the first equation, y=12,000−xy = 12,000 - x. Substituting this into the second equation gives 2x+3(12,000−x)=29,0002x + 3(12,000 - x) = 29,000. Distributing the 33 yields 2x+36,000−3x=29,0002x + 36,000 - 3x = 29,000. Combining like terms results in −x+36,000=29,000-x + 36,000 = 29,000. Subtracting 36,00036,000 from both sides gives −x=−7,000-x = -7,000, which simplifies to x=7,000x = 7,000.

Key Concept

Solving systems of linear equations using substitution or elimination methods
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Topics

Systems of Linear Equations

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