Solving Problems Involving Linear Functions

||y=ax+b|| where
|a:| rate of change
|b:| |y|-intercept

1. Determine the independent variable and the dependent variable.

2. Find the rule.

   1. Find parameter a.

   2. Find parameter b.

3. Replace the given value into the rule.

4. Isolate the other variable.

5. Answer the question.

• To visualize the situation, graph the function on a Cartesian plane or sketch the situation. Although the graph is not accurate enough to find the answer, it enables you to check whether the data found works in the context.

• Sometimes the rate of change and the initial value are given directly in the context of the problem. In this case, it is not necessary to calculate them.

Eric has just found a new job as a strawberry picker on Île d'Orléans. On his first day, he discovers that the previous picker left him a number of baskets full of strawberries. At the end of the day, Eric has a total of |31| full baskets.

Knowing that Eric works |6| hours a day and can fill |3| baskets in |1| hour, answer the following questions.

Question 1: What is the total number of strawberry baskets Eric will fill after |5| days of work?
Question 2: After how many days of work will he fill a total of |157| baskets?

1. Determine the independent variable and the dependent variable
   We're looking for the quantity of filled strawberry baskets regarding the time.
   |x:| time (hours)
   |y:| number of filled strawberry baskets

2. Find the rule

   1. Find parameter |\boldsymbol{a}|
      The problem states that Eric fills |3| baskets every hour. This is the rate of change.
      ||a=3||

   2. Find parameter |\boldsymbol{b}|
      The |y|-intercept is the initial value. In context, this is the quantity of strawberry baskets filled before Eric started picking. To find the quantity, we replace the parameter |a| with |3| in the rule and replace |x| and |y| with a given point. After his first day, there are |31| baskets. He works |6| hours per day, therefore, we can determine that |(6,31)| is a point.
      ||\begin{align}y&=ax+b\\y&=3x+b\\\\\color{#3a9a38}{31}&=3(\color{#3a9a38}6)+b\\31&=18+b\\31\color{#ec0000}{-18}&=18+b\color{#ec0000}{-18}\\13&=b\end{align}||
      The rule of the function is therefore |y=3x+13.|

Question 1: What is the total number of strawberry baskets filled after |5| days of work?

3. Replace the given value into the rule
   The given value is a time, so it is an |x|-value. We must therefore replace |x| in the rule and find the corresponding |y|-value. However, first we must convert |5| days to hours, since the measurement units for the rate of change are |\text{baskets/hour}.| Since Eric works |6| hours per day, |5| days represent |5\times6=30| hours of work. We therefore replace |x| by |30.|
   ||y=3(30)+13||

4. Isolate the other variable
   Since |y| is already isolated, we simply complete the calculation.
   ||\begin{align}y&=3\times30+13\\&=90+13\\&=103\end{align}||

5. Answer the question
   After |5| days of work, there will be a total of |103| strawberry baskets filled.

Question 2: After how many days of work will he have filled a total of |\boldsymbol{157}| baskets?

3. Replace the given value into the rule
   The given value is a number of baskets, so a |y|-value. Replace |y| in the rule and find the corresponding |x|-value.
   ||157=3x+13||

4. Isolate the other variable
   ||\begin{align}157&=3x+13\\157\color{#ec0000}{-13}&=3x+13\color{#ec0000}{-13}\\144&=3x\\\color{#ec0000}{\dfrac{\color{black}{144}}{3}}&=\color{#ec0000}{\dfrac{\color{black}{3x}}{3}}\\48&=x\end{align}||

5. Answer the question
   We know that after |48| hours of work, there will be |157| baskets. However, the question asks for a number of days. So we must divide the number of hours by |6,| since Eric works |6| hours per day. ||48\ \text{hours}\div6\ \text{hours/day}=8\ \text{days}|| Therefore, there will be |157| baskets of strawberries filled after |8| days of work.

Henry is a window cleaner. A company wants to hire him to clean office windows on a downtown skyscraper. They offer him 2 remuneration options.

1. A flat rate of |$2\ 205,| regardless of the number of windows cleaned.

2. A base amount of |$500| with an extra |$2.75| per window cleaned.

Given that the company’s skyscraper has |593| windows, answer the following questions.

Question 1: Which option is best for Henry?
Question 2: How many windows must Henry clean for the 2 options to be equivalent?

Amina and Gabrielle enjoy reading. They decide to read the same novel at the same time for fun. When they get together to talk about the book for the first time, Amina has already read |50| pages at a rate of |28| pages every |5| days. Gabrielle has read |36| pages and reached |78| pages after |6| days. Let’s assume the reading pace of the 2 friends does not change over the course of the book. Answer the following questions.

Question 1: If the book has |246| pages, who will finish it first?
Question 2: When one friend finishes the book, how many pages will the other friend have left to read?