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Sagot :
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]
#### 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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