Question

WRITING SIMULTANEOUS LINEAR EQUATIONS QUICK CHECK:
USE THE TABLE TO ANSWER THE QUESTION.
TABLE OF VALUES
X YA YB
0 0 3
1 2 3.5
2 4 4
WHICH OF THE FOLLOWING SYSTEMS OF LINEAR EQUATIONS DOES THE TABLE OF VALUES REPRESENT?

1. Y = 2X
Y = 3X + 1/2
2. Y = 2X + 3
Y = 1/2 X
3. Y = 2 X
Y = 1/2X + 3
4. Y = 1/2 X
Y = 2X + 3

Answer 1

To determine which system of linear equations the table represents, we need to find the equations for both \( Y_A \) and \( Y_B \) based on the provided values in the table.

### Table of Values
```
X | YA | YB
0 | 0 | 3
1 | 2 | 3.5
2 | 4 | 4
```

### For \( Y_A \):
- \( Y_A \) values are (0, 0), (1, 2), (2, 4).
- The pattern shows that \( Y_A \) increases by 2 when \( X \) increases by 1.
- Thus, we can see that:
\[ Y_A = 2X \]

### For \( Y_B \):
- \( Y_B \) values are (0, 3), (1, 3.5), (2, 4).
- Let's calculate the slope between the points:
1. From (0, 3) to (1, 3.5):
\[
\text{slope} = \frac{3.5 - 3}{1 - 0} = 0.5
\]
2. From (1, 3.5) to (2, 4):
\[
\text{slope} = \frac{4 - 3.5}{2 - 1} = 0.5
\]
- This indicates a slope of \( \frac{1}{2} \), so we can express \( Y_B \) as:
\[ Y_B = \frac{1}{2}X + 3 \] (We use the point (0, 3) to find the y-intercept; when \( X = 0 \), \( Y_B = 3 \))

### Summary of the equations:
- \( Y_A = 2X \)
- \( Y_B = \frac{1}{2}X + 3 \)

### Matching with the provided options:
1. \( Y = 2X \) \ \ and \ \ \( Y = 3X + \frac{1}{2} \) (Not a match)
2. \( Y = 2X + 3 \) \ and \ \ \( Y = \frac{1}{2}X \) (Not a match)
3. \( Y = 2X \) \ and \ \ \( Y = \frac{1}{2}X + 3 \) (This is a match!)
4. \( Y = \frac{1}{2}X \) \ and \ \ \( Y = 2X + 3 \) (Not a match)

### Conclusion:
The system of linear equations represented by the table of values is **Option 3**:
\[ Y = 2X \quad \text{and} \quad Y = \frac{1}{2}X + 3 \]