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Part 1: Matchsticks

At home, John made the following matchstick pattern using matchsticks from his pantry. John created a shape using 5 matchsticks. He then decided to add three more to his second shape.

1
2
(6 Marks)

1. Complete the following table using the matchsticks above:
[tex]\[
\begin{tabular}{|l|c|c|c|}
\hline
\begin{tabular}{l}
Pattern \\
Number $(n):$
\end{tabular} & 1 & 2 & 3 \\
\hline
\begin{tabular}{l}
Number of \\
Matchsticks $(P):$
\end{tabular} & 5 & 8 & \\
\hline
\end{tabular}
\][/tex]

Each time he added to his shape, he noticed there was a pattern occurring. John discovered that he could create a rule. He recalled from his Mathematics class that:

a. Rewrite the linear rule above \([tex]y=mx+c[tex]\), using the variables of \([tex]n[tex]\) (Pattern Number) and [tex]\([tex]P[tex]\)[/tex] (Number of Matchsticks).
[tex]\[
P = mn + c
\][/tex]

b. John recalled that the \([tex]m[tex]\) value was the change in the \([tex]y[tex]\) values. Find the gradient [tex]\([tex]m[tex]\)[/tex] of the matchstick pattern:
[tex]\[
m = 3
\][/tex]

c. The \([tex]c[tex]\) value is when \([tex]x=0[tex]\). In John's equation, it is when [tex]\([tex]n=0[tex]\)[/tex]. Determine the \([tex]c[tex]\) value for this equation:
[tex]\[
c = 2
\][/tex]

d. Help John to rewrite his linear equation, using his newfound \([tex]m[tex]\) and \([tex]c[tex]\) values:
[tex]\[
P = 3n + 2
\][/tex]

Sagot :

GinnyAnsweridn
Sure! Let's tackle each part of the question step-by-step.

#### Part 1
1. Completing the Table:
Let's analyze the given information and fill out the table.
- Pattern 1 has 5 matchsticks.
- For Pattern 2, John added 3 more matchsticks.

Given:
- Pattern 1: 5 matchsticks
- Pattern 2: 5 + 3 = 8 matchsticks
- Pattern 3: 8 + 3 = 11 matchsticks

So, the table should be:

[tex]\[ \begin{tabular}{|l|c|c|c|} \hline \begin{tabular}{l} Pattern \\ Number $(n):$ \end{tabular} & 1 & 2 & 3 \\ \hline \begin{tabular}{l} Number of \\ Matchsticks $(P):$ \end{tabular} & 5 & 8 & 11 \\ \hline \end{tabular} \][/tex]

#### Part 2
a. Rewriting the Linear Rule:
In a linear equation, the standard form is [tex]\( y = mx + c \)[/tex].

Using the variables given:
- [tex]\( n \)[/tex] = Pattern Number
- [tex]\( P \)[/tex] = Number of Matchsticks

The linear rule should be:
[tex]\[ P = mn + c \][/tex]

b. Finding the Gradient (m):
The gradient [tex]\( m \)[/tex] is the change in [tex]\( y \)[/tex] (Number of Matchsticks) per change in [tex]\( x \)[/tex] (Pattern Number).

From Pattern 1 to 2:
[tex]\[ \Delta P = 8 - 5 = 3 \][/tex]
[tex]\[ \Delta n = 2 - 1 = 1 \][/tex]

Therefore, the gradient [tex]\( m \)[/tex] is:
[tex]\[ m = \frac{\Delta P}{\Delta n} = \frac{3}{1} = 3 \][/tex]

So, [tex]\( m = 3 \)[/tex].

c. Determining the Constant (c):
The constant [tex]\( c \)[/tex] is the value of [tex]\( P \)[/tex] when [tex]\( n = 0 \)[/tex].

From the equation:
[tex]\[ P = mn + c \][/tex]

Using the first pattern (n = 1, P = 5):
[tex]\[ 5 = 3 \cdot 1 + c \][/tex]
[tex]\[ 5 = 3 + c \][/tex]
[tex]\[ c = 5 - 3 \][/tex]
[tex]\[ c = 2 \][/tex]

So, [tex]\( c = 2 \)[/tex].

d. Rewriting the Linear Equation:
Now that we have the values of [tex]\( m \)[/tex] and [tex]\( c \)[/tex], we can rewrite the equation:

[tex]\[ P = 3n + 2 \][/tex]

This linear equation represents the relationship between the Pattern Number [tex]\( n \)[/tex] and the Number of Matchsticks [tex]\( P \)[/tex].

To summarize:

1. The complete table is:
[tex]\[ \begin{tabular}{|l|c|c|c|} \hline \begin{tabular}{l} Pattern \\ Number $(n):$ \end{tabular} & 1 & 2 & 3 \\ \hline \begin{tabular}{l} Number of \\ Matchsticks $(P):$ \end{tabular} & 5 & 8 & 11 \\ \hline \end{tabular} \][/tex]

2. The rewritten linear rule is: [tex]\( P = mn + c \)[/tex]

3. The gradient [tex]\( m \)[/tex] is 3.

4. The constant [tex]\( c \)[/tex] is 2.

5. The final linear equation is: [tex]\( P = 3n + 2 \)[/tex]
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