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=2xy=3x+12


{y=12xy=2x+3


{y=2x+3y=12x

Answer 1

To find which system of linear equations the table of values represents, we need to analyze the points given in the table for \(y_a\) and \(y_b\) based on their corresponding \(x\) values.

From the table provided:

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

This suggests a linear relationship. Let's see if we can express it as a linear equation in the form \(y = mx + b\):
- The slope \(m\) can be calculated using two points, for example, the points (0, 0) and (1, 2):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{1 - 0} = 2
\]

Using this slope and the point (0, 0) to find the y-intercept \(b\):
\[
y_a = 2x + 0 \implies y_a = 2x
\]

So the equation for \(y_a\) is:
\[
y_a = 2x
\]

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

Let's find the slope between the points (0, 3) and (1, 3.5):
\[
m = \frac{3.5 - 3}{1 - 0} = \frac{0.5}{1} = 0.5
\]

Using this slope and the point (0, 3) to find the intercept \(b\):
\[
y_b = 0.5x + 3
\]

So the equation for \(y_b\) is:
\[
y_b = 0.5x + 3
\]

Now let's write the system of equations we've found:
1. \(y_a = 2x\)
2. \(y_b = 0.5x + 3\)

None of the provided response formats exactly match our derived equations directly, but they can be manipulated into a comparable form. The options given are:

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

None of the provided options exactly correspond with our equations. However, this can be somewhat confusing due to incorrect formatting of the equations in the options.

Based on the analysis, we expect a system representing \(y_a = 2x\) and a similar linear function for \(y_b\) which does not exactly fit the available options provided but rather appears more similar to structure of y in the options.

Given that this analysis is to find one closest possible option with \(y_a\) yielding a direct \(2x\), it would appear that \(y = 2x\) is correct, and in terms of something like \(y_b\), consider those coefficients appropriately. Please ensure to check multiple times for expected relationship in systematic representation based on format processing and that denotes moving statistically toward these provided.