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Algebra Difficulty: Medium

A linear model estimates the population of a city from $1991$ to $2015$ . The model estimates the population was $57$ thousand in $1991$ , $224$ thousand in $2011$ , and $x$ thousand in $2015$ . To the nearest whole number, what is the value of $x$ ?

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Explanation

The correct answer is $257$ . It’s given that a linear model estimates the population of a city from $1991$ to $2015$ . Since the population can be estimated using a linear model, it follows that there is a constant rate of change for the model. It’s also given that the model estimates the population was $57$ thousand in $1991$ , $224$ thousand in $2011$ , and $x$ thousand in $2015$ . The change in the population between $2011$ and $1991$ is $224 minus 57$ , or $167$ , thousand. The change in the number of years between $2011$ and $1991$ is $2011 minus 1991$ , or $20$ , years. Dividing $167$ by $20$ gives $167 divided by 20$ , or $8.35$ , thousand per year. Thus, the change in population per year from $1991$ to $2015$ estimated by the model is $8.35$ thousand. The change in the number of years between $2015$ and $2011$ is $2015 minus 2011$ , or $4$ , years. Multiplying the change in population per year by the change in number of years yields the increase in population from $2011$ to $2015$ estimated by the model: $left parenthesis 8.35 right parenthesis left parenthesis 4 right parenthesis$ , or $33.4$ , thousand. Adding the change in population from $2011$ to $2015$ estimated by the model to the estimated population in $2011$ yields the estimated population in $2015$ . Thus, the estimated population in $2015$ is $33.4 plus 224$ , or $257.4$ , thousand. Therefore to the nearest whole number, the value of $x$ is $257$ .