Answer :
Let's solve the problem step-by-step. We will complete the table using the given starting numbers and rules for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] patterns.
### Step-by-Step Solution:
#### Step 1: Understand the patterns
1. Pattern [tex]\(x\)[/tex]:
- Starting number: 3
- Rule: Add 3 each time
2. Pattern [tex]\(y\)[/tex]:
- Starting number: 6
- Rule: Add 6 each time
#### Step 2: Generate the [tex]\(x\)[/tex]-values using Pattern [tex]\(x\)[/tex]
- Starting number is 3.
- Following the rule of adding 3 each time:
1. First value: [tex]\(3\)[/tex]
2. Second value: [tex]\(3 + 3 = 6\)[/tex]
3. Third value: [tex]\(6 + 3 = 9\)[/tex]
Thus, the [tex]\(x\)[/tex]-values are [tex]\(3, 6, 9\)[/tex].
#### Step 3: Generate the [tex]\(y\)[/tex]-values using Pattern [tex]\(y\)[/tex]
- Starting number is 6.
- Following the rule of adding 6 each time:
1. First value: [tex]\(6\)[/tex]
2. Second value: [tex]\(6 + 6 = 12\)[/tex]
3. Third value: [tex]\(12 + 6 = 18\)[/tex]
Thus, the [tex]\(y\)[/tex]-values are [tex]\(6, 12, 18\)[/tex].
#### Step 4: Complete the table
Filling in the values derived from patterns:
[tex]\[ \begin{array}{cc} x & y \\ \hline 3 & 6 \\ \hline 6 & 12 \\ \hline 9 & 18 \\ \hline \end{array} \][/tex]
### Step 5: Plot the ordered pairs [tex]\((x, y)\)[/tex] on the graph
The ordered pairs from the table are:
- [tex]\((3, 6)\)[/tex]
- [tex]\((6, 12)\)[/tex]
- [tex]\((9, 18)\)[/tex]
With these pairs, plot each point on the coordinate plane:
1. Plot the point [tex]\((3, 6)\)[/tex]: Move 3 units to the right on the [tex]\(x\)[/tex]-axis and 6 units up on the [tex]\(y\)[/tex]-axis.
2. Plot the point [tex]\((6, 12)\)[/tex]: Move 6 units to the right on the [tex]\(x\)[/tex]-axis and 12 units up on the [tex]\(y\)[/tex]-axis.
3. Plot the point [tex]\((9, 18)\)[/tex]: Move 9 units to the right on the [tex]\(x\)[/tex]-axis and 18 units up on the [tex]\(y\)[/tex]-axis.
The resulting graph will have these three points plotted, which should lie in a straight line given the constant addition rules for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
### Step-by-Step Solution:
#### Step 1: Understand the patterns
1. Pattern [tex]\(x\)[/tex]:
- Starting number: 3
- Rule: Add 3 each time
2. Pattern [tex]\(y\)[/tex]:
- Starting number: 6
- Rule: Add 6 each time
#### Step 2: Generate the [tex]\(x\)[/tex]-values using Pattern [tex]\(x\)[/tex]
- Starting number is 3.
- Following the rule of adding 3 each time:
1. First value: [tex]\(3\)[/tex]
2. Second value: [tex]\(3 + 3 = 6\)[/tex]
3. Third value: [tex]\(6 + 3 = 9\)[/tex]
Thus, the [tex]\(x\)[/tex]-values are [tex]\(3, 6, 9\)[/tex].
#### Step 3: Generate the [tex]\(y\)[/tex]-values using Pattern [tex]\(y\)[/tex]
- Starting number is 6.
- Following the rule of adding 6 each time:
1. First value: [tex]\(6\)[/tex]
2. Second value: [tex]\(6 + 6 = 12\)[/tex]
3. Third value: [tex]\(12 + 6 = 18\)[/tex]
Thus, the [tex]\(y\)[/tex]-values are [tex]\(6, 12, 18\)[/tex].
#### Step 4: Complete the table
Filling in the values derived from patterns:
[tex]\[ \begin{array}{cc} x & y \\ \hline 3 & 6 \\ \hline 6 & 12 \\ \hline 9 & 18 \\ \hline \end{array} \][/tex]
### Step 5: Plot the ordered pairs [tex]\((x, y)\)[/tex] on the graph
The ordered pairs from the table are:
- [tex]\((3, 6)\)[/tex]
- [tex]\((6, 12)\)[/tex]
- [tex]\((9, 18)\)[/tex]
With these pairs, plot each point on the coordinate plane:
1. Plot the point [tex]\((3, 6)\)[/tex]: Move 3 units to the right on the [tex]\(x\)[/tex]-axis and 6 units up on the [tex]\(y\)[/tex]-axis.
2. Plot the point [tex]\((6, 12)\)[/tex]: Move 6 units to the right on the [tex]\(x\)[/tex]-axis and 12 units up on the [tex]\(y\)[/tex]-axis.
3. Plot the point [tex]\((9, 18)\)[/tex]: Move 9 units to the right on the [tex]\(x\)[/tex]-axis and 18 units up on the [tex]\(y\)[/tex]-axis.
The resulting graph will have these three points plotted, which should lie in a straight line given the constant addition rules for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
