Question

Use the table to answer the question.

Table of Values

x
ya yb
0 3 18
1 6 1312
2 9 9
Which of the following systems of linear equations does the table of values represent?

(1 point)
Responses

{y=−3x+3y=412x+18

{y=3x+3y=412x+18

{y=3x+3y=412x−18

{y=3x+3y=−412x+18
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Answer 1

To determine which system of linear equations the table of values represents, we can find the equations from the given coordinates for \( y_a \) and \( y_b \).

For \( y_a \):

- When \( x = 0, y_a = 3 \)
- When \( x = 1, y_a = 6 \)
- When \( x = 2, y_a = 9 \)

From these points, we can find the slope (m):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 3}{1 - 0} = 3
\]

Using the point-slope form \( y - y_1 = m(x - x_1) \) with the point (0, 3):

\[
y - 3 = 3(x - 0) \rightarrow y = 3x + 3
\]

So, one equation is:

\[
y = 3x + 3 \quad (1)
\]

For \( y_b \):

- When \( x = 0, y_b = 18 \)
- When \( x = 1, y_b = 1312 \)
- When \( x = 2, y_b = 9 \)

The points seem inconsistent, but we can analyze the first two points to find the slope for the equation \( y_b \):

Taking the difference between Points (0, 18) and (1, 1312):

\[
m = \frac{1312 - 18}{1 - 0} = 1294
\]

Using point-slope form with (0, 18):

\[
y - 18 = 1294(x - 0) \rightarrow y = 1294x + 18
\]

However, this doesn't fit neatly into the options provided. Let's analyze the other points as well.

Given the proximity to linear behavior and \( y_b \) doesn't look like it represents a simple linear equation from our analysis.

Now, potentially included systems from the responses are:

1. \( y = -3x + 3 \)
2. \( y = 3x + 3 \)
3. \( y = 3x - 18 \)
4. \( y = 3x - 4 \)

From the analysis, we see the first equation for \( y_a = 3x + 3 \) matches:

So we know \( y_a = 3x + 3 \).

The closest option for \( y_a \) based on our analysis appears to be straightforward from the given data.

So analyzing against the options, the system described in one of the responses is likely:

\(
\{y=3x+3 \quad; \quad y=1294x+18\}
\)

However, for completeness, we might not have \( y_b \) distinctly aligned by given original linear equations. The linear equation for \( y_a \) stands out.

The correct answer to the options you provided is:

\(
{y=3x+3 \quad} (the other equation representation needs clarity which might not be directly pulling linear)
\)

If the displayed options contort from \( y_a \), then you could opt for:

**\( {y=3x+3 \; y = 4 \; 12x+18} \)** if they aligned with a potential linear regression (keeping context). If parameters aren't stable here, it would warrant examination.

In summary, the obvious pick begins with:

\(
{y=3x+3}
\)

Ultimately, adjust closure with exactly what fits align with \( y_a, y_b \).