Use the table to answer the question.

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=2xy=12x+3

{y=2x+3y=12x


{y=12xy=2x+3


{y=2xy=3x+12

Answer 1

To determine which system of linear equations the table represents, we need to analyze the values for \(y_a\) and \(y_b\) based on the provided \(x\) values:

Given table:

x   ya  yb
0   0   3
1   2   3.5
2   4   4

Analyzing \(y_a\):

Let's find the equation for \(y_a\):

• For \(x = 0\), \(y_a = 0\)
• For \(x = 1\), \(y_a = 2\)
• For \(x = 2\), \(y_a = 4\)

Observing these values, we see that as \(x\) increases by 1, \(y_a\) increases by 2. This suggests a linear relationship with the form: \[ y_a = 2x \] To confirm:

• When \(x = 0\), \(y_a = 2(0) = 0\) (correct)
• When \(x = 1\), \(y_a = 2(1) = 2\) (correct)
• When \(x = 2\), \(y_a = 2(2) = 4\) (correct)

Analyzing \(y_b\):

Now, let's find the equation for \(y_b\):

• For \(x = 0\), \(y_b = 3\)
• For \(x = 1\), \(y_b = 3.5\)
• For \(x = 2\), \(y_b = 4\)

This indicates an increase of 0.5 for each increment of \(x\). Hence, it suggests a linear relationship: \[ y_b = 3 + 0.5x \] To confirm:

• When \(x = 0\), \(y_b = 3 + 0.5(0) = 3\) (correct)
• When \(x = 1\), \(y_b = 3 + 0.5(1) = 3.5\) (correct)
• When \(x = 2\), \(y_b = 3 + 0.5(2) = 4\) (correct)

System of Linear Equations:

Now that we have determined:

• \(y_a = 2x\)
• \(y_b = 3 + 0.5x\)

Let's check the options given:

1. \(y = 2x, y = 12x + 3\) - This doesn't match.
2. \(y = 12x, y = 2x + 3\) - This doesn't match.
3. \(y = \frac{1}{2}x, y = 2x + 3\) - This doesn't match.
4. \(y = 2x, y = 3x + 12\) - This doesn't match.

None of the options provided exactly reflects the determined equations. However, it seems like the closest set is likely option 1 after interpreting and adjusting it.

Hence, if the answer must correlate strictly with values, the "closest" could relate back based on basic misinterpret expectations as no other further breakdown will specifically confirm except for something we showcased. Thus, none matches precisely, possibly signifying that if answers must stick, it locks into the slope more so than pure identity response.

Thus, from our analysis:

The matched response most inherently recognized fell on option closest to:

1. \(y = 2x, y = 12x + 3\), by that definition translated.