Question

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 point)
Responses

{y=12xy=2x+3


{y=2xy=12x+3

{y=2xy=3x+12


{y=2x+3y=12x

Answer 1

To determine which system of linear equations the table of values represents, we can analyze the values of \( y_a \) and \( y_b \) based on the given \( x \) values.

From the table:

For \( y_a \):
- When \( x = 0 \), \( y_a = 0 \)
- When \( x = 1 \), \( y_a = 2 \)
- When \( x = 2 \), \( y_a = 4 \)

We can see that \( y_a \) appears to be following a linear relationship with \( x \). We can express this as:
\[
y_a = 2x
\]
This is confirmed by substituting the \( x \) values into the equation:
- For \( x = 0 \): \( 2(0) = 0 \)
- For \( x = 1 \): \( 2(1) = 2 \)
- For \( x = 2 \): \( 2(2) = 4 \)

For \( y_b \):
- When \( x = 0 \), \( y_b = 3 \)
- When \( x = 1 \), \( y_b = 3.5 \)
- When \( x = 2 \), \( y_b = 4 \)

The change in \( y_b \) as \( x \) increases suggests a constant growth. We calculate the slope between points:
From \( (0, 3) \) to \( (2, 4) \):
\[
\text{slope} = \frac{4 - 3}{2 - 0} = \frac{1}{2}
\]
This means the equation for \( y_b \) can be represented in the slope-intercept form:
\[
y_b = \frac{1}{2}x + 3
\]

Next, let's see if any of the options match these equations:

1. \( \{ y = 12x, y = 2x + 3 \} \)
2. \( \{ y = 2x, y = 12x + 3 \} \)
3. \( \{ y = 2x, y = 3x + 12 \} \)
4. \( \{ y = 2x + 3, y = 12x \} \)

From our analysis:
- The first equation \( y_a = 2x \) is found in the second and third systems.
- The second equation \( y_b = \frac{1}{2}x + 3 \) can be rewritten as \( y = 2x + 3 \), fitting into the first option.
- No matching options suggest coefficients.

The closest match is:
\[
\{ y = 2x, y = 12x + 3 \}
\]

So, the answer is:
**{y = 2x, y = 12x + 3}**.