# Problem Solving With Linear Functions Key

We followed the steps of modeling a problem to analyze the information.

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A negative input value could refer to a number of weeks before she saved $3,500, but the scenario discussed poses the question once she saved$3,500 because this is when her trip and subsequent spending starts.

It is also likely that this model is not valid after the x-intercept, unless Emily will use a credit card and goes into debt.

The amount of money she has remaining while on vacation depends on how long she stays.

We can use this information to define our variables, including units.

The domain represents the set of input values, so the reasonable domain for this function is $$0t8.75$$.

In the above example, we were given a written description of the situation.Rate of Change: She anticipates spending 0 each week, so –0 per week is the rate of change, or slope.Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. The problem should list the Y- intercept, a starting amount of something and a slope, or a rate of change. You can tell that you need to create a linear equation by the information the problem gives you.Identify a solution pathway from the provided information to what we are trying to find. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem. Clearly convey your result using appropriate units, and answer in full sentences when necessary. In her situation, there are two changing quantities: time and money.To find the x-intercept, we set the output to zero, and solve for the input.$\begin 0&=−400t 3500 \ t&=\dfrac \ &=8.75 \end$ The x-intercept is 8.75 weeks.Also, because the slope is negative, the linear function is decreasing.This should make sense because she is spending money each week.

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