Linear Functions(1 point)

Question

1. Tommy the cat is constantly stealing hair ties. At the start of the week, he had already stolen 5 hair ties. Each day, Tommy steals 3 more hair ties.

Part A: Write a linear equation that represents the total number of hair ties (h) based on the number of days that have passed (d).

h(d)=
ℎ
(
𝑑
)
=


Part B: Tommy would have stolen 32 hair ties after 9 days.



2. Penny earned a monetary reward for returning a lost puppy. She decides to spend $12 from the reward each week on treats for her own pets. After 4 weeks Penny has $134 left.

Part A: What is the slope and what is one point that we know?

m=
𝑚
=

a point that we know is (write your response as an ordered pair)
Part B: Write a linear equation that models the total money left, y, from her reward after x weeks.

y=
𝑦
=
(Write your answer in slope y-intercept form:y=mx+b
𝑦
=
𝑚
𝑥
+
𝑏
)

Answer 1

Sure! Let's work through your questions step by step.

Problem 1: Tommy the Cat and Hair Ties

Part A: Write a linear equation.

Tommy starts the week with 5 hair ties and steals 3 more each day. Therefore, the total number of hair ties \( h \) as a function of the number of days \( d \) is given by:

\[ h(d) = 3d + 5 \]

Part B: Validate the statement that Tommy would have stolen 32 hair ties after 9 days.

We can further analyze this by substituting \( d = 9 \) into the function.

\[ h(9) = 3(9) + 5 = 27 + 5 = 32 \]

So, after 9 days, Tommy indeed would have stolen 32 hair ties.

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Problem 2: Penny and Her Monetary Reward

Part A: Determine the slope and known point.

1. Slope (m): Penny spends $12 each week, which indicates her spending rate is $-12 per week. Thus: \[ m = -12 \]

2. Known Point: After 4 weeks, she has $134 left. This gives us the point: \[ (4, 134) \]

Part B: Write a linear equation that models the total money left.

Using the slope-intercept form \( y = mx + b \):

We already have \( m = -12 \) and we can use the point (4, 134) to find \( b \).

Substituting into the equation: \[ 134 = -12(4) + b \] Calculating: \[ 134 = -48 + b \] \[ b = 134 + 48 = 182 \]

So the linear equation that models the total money left \( y \) from her reward after \( x \) weeks is:

\[ y = -12x + 182 \]

To summarize:

1. Tommy the cat's hair ties:

   • Linear equation: \( h(d) = 3d + 5 \)

2. Penny's monetary reward:

   • Slope: \( m = -12 \)
   • Known point: \( (4, 134) \)
   • Linear equation: \( y = -12x + 182 \)