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In a certain region, about 6% of a city’s population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2015, there were 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where $\mathbf{x}_{0}$ is the initial population in 2015 . Then estimate the populations in the city and in the suburbs two years later, in 2017 .

(a) $\left[\begin{array}{ll}{0.94} & {0.04} \\ {0.06} & {0.96}\end{array}\right] \mathbf{x}_{k}=\mathbf{x}_{k+1}$(b) $\left[\begin{array}{l}{8920800} \\ {1879200}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 10

Linear Models in Business, Science, and Engineering

Introduction to Matrices

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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Okay, So in this problem, um, we want to set up a difference equation. Except K plus one equals two. A tradition may metrics and times, except K. I'm so for each x sub k. The first entrance, the population in the city, the second change the population in the surrounding area and the full transition metrics the engine will be so because we defied our, um, they were bring this way. So for the transition metrics on the frosting trim, we will be the people stay, you see. And the second you will be the people move from the surrounding area. Two of those et the 30 into it will be the people move from the city to the surrounding area. The last intern will be the people, um, stay Penis around the area. Okay, so you if we if we understand this, those legit geometrics in this case will be your night. 0.94 You're going 04 to open 06 and the 0.96 So this state eyes are given in the problem. We just need toe Rearrange this in the correct order in the initial state eg zero will be, um so that would be 10 million and uh, 100,000. Okay, so we are looking for, um after two years, with the population in the city and the water population in the surrounding area, that would be x two extra, because to end times X one, which is in terms in tans x zero. So bad calculator you will be a foreign result. So this is the solution to this.

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