Question

Table of values:
x ya yb
0 3 18
1 6 13 1/2
2 9 9

Which of the following systems of linear equations does the table of values represent?

A: {y = -3x + 3
{y = 41/2x + 18

B: { y= 3x + 3
{ y = 41/2x - 18

C: { y = 3x + 3
{ y = -41/2x + 18

D: { y = 3x + 3
{ y = 41/2x + 18

Answer 1

To determine which system of linear equations corresponds to the table of values provided, we need to analyze the values of \( y_a \) and \( y_b \) for each value of \( x \).

### Analyzing \( y_a \)

1. For \( x = 0 \): \( y_a = 3 \)
2. For \( x = 1 \): \( y_a = 6 \)
3. For \( x = 2 \): \( y_a = 9 \)

We can see there is a consistent increase:
- From \( 3 \) to \( 6 \): an increase of \( 3 \) (which is \( 3 \times 1 \))
- From \( 6 \) to \( 9 \): an increase of \( 3 \) (which is \( 3 \times 1 \))

This indicates that \( y_a \) follows the equation \( y = 3x + 3 \).

### Analyzing \( y_b \)

1. For \( x = 0 \): \( y_b = 18 \)
2. For \( x = 1 \): \( y_b = 13.5 \) or \( y_b = \frac{27}{2} \)
3. For \( x = 2 \): \( y_b = 9 \)

To find the slope:
- From \( 18 \) to \( 13.5 \): The decrease is \( 18 - 13.5 = 4.5 \) (which occurs over an interval of \( 1 \)), giving us a slope of \( -4.5 \).
- From \( 13.5 \) to \( 9 \): The decrease is \( 13.5 - 9 = 4.5 \) again.

So the slope of the line can be calculated as:
\[
\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 18}{2 - 0} = \frac{-9}{2} = -\frac{9/2}{1}
\]

Finding the equation:
Using the point \( (0, 18) \):
\[
y - 18 = -\frac{9}{2}(x - 0)
\]
Therefore, the equation becomes:
\[
y = -\frac{9}{2}x + 18
\]

### Identifying the System of Equations

Considering the equations we've derived:
- \( y_a \) is represented by \( y = 3x + 3 \).
- \( y_b \) is represented by \( y = -\frac{9}{2}x + 18 \) which is incorrect according to given options which have a positive fraction.

Recomputing it again:
For \( x = 0 \) we have \( y_b = 18 = b \)
For \( x = 1 \) we see the change in \( y \): from 18 to 13.5 = -4.5 and slope is -4.5.
Combining them we realize it means it follows the linearity thus:
When we put it all together we notice indeed it could be approximated by negative, resulting in new slope considerations.

**Final solution:**
Looking into the options provided:
- Since \( y_a \) is correctly \( 3x +3 \) and as \( y_b \) is sliding approximately but needs reviewing implies inclination with valid patterns following linear regress to confirm where other options hold strong.

After comparing:
We can assert that:

### Final Selection:
The valid system of equations from the options given is:

**C:**
\[
{ y = 3x + 3 }
\]
\[
{ y = -\frac{41}{2}x + 18 }
\]

Correctly noting structure.