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Use the table to answer the question.

Table of Values

x
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
=
3
x
+
3
y
=
−
4
1
2
x
+
18

{y=−3x+3y=412x+18
{
y
=
−
3
x
+
3
y
=
4
1
2
x
+
18

{y=3x+3y=412x−18
{
y
=
3
x
+
3
y
=
4
1
2
x
−
18

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

3 answers

To determine which system of linear equations corresponds to the provided table of values, we will analyze the data and try to derive equations for \( y_a \) and \( y_b \) based on the given \( x \) values.

Given Table of Values:

| x | y_a | y_b | |---|-----|-----| | 0 | 3 | 18 | | 1 | 6 | 13 | | 2 | 9 | 9 |

Analysis for \( y_a \):

  1. We notice that for \( y_a \):
    • At \( x = 0 \), \( y_a = 3 \)
    • At \( x = 1 \), \( y_a = 6 \)
    • At \( x = 2 \), \( y_a = 9 \)

The increment of \( y_a \) as \( x \) increases is consistent:

  • \( 6 - 3 = 3 \)
  • \( 9 - 6 = 3 \)

This indicates a linear relationship with the slope of 3. The equation can be formulated as: \[ y_a = 3x + 3 \]

Analysis for \( y_b \):

  1. Now let's analyze \( y_b \):
    • At \( x = 0 \), \( y_b = 18 \)
    • At \( x = 1 \), \( y_b = 13 \)
    • At \( x = 2 \), \( y_b = 9 \)

Here, we notice that as \( x \) increases, \( y_b \) decreases:

  • \( 13 - 18 = -5 \)
  • \( 9 - 13 = -4 \)

This indicates a linear relationship but with a different slope. Let's check its equation:

  • The first difference is \(-5\) and the second difference is \(-4\).

To find the pattern: Starting from \( 18 \) at \( x = 0 \): \[ y_b = 18 - nx \] Using the first two data points to derive \( y_b \):

  • \( y_b = -5x + 18 \) However, the data doesn’t seem to perfectly fit a single linear model.

Constructing the Equations:

So we have derived the following equations based on our analysis:

  • \( y_a = 3x + 3 \)
  • \( y_b = -5x + 18 \)

Conclusion:

Looking at the options provided, we need to find a system of equations consistent with these forms. The options you provided do not exactly contain the equations we derived, but let's match what we see most closely: Upon review, it appears from the options, the one which is suggested for \( y_a \) closely resembles \( y = 3x + 3 \), suggesting that for \( y_b \), somewhat higher values, identified with levels or differences around \( 4 \).

Given options do not mirror derived equations accurately. Choices may need review or imply simplifications not described.

Thus, it seems the right solution is: \[ { y = 3x + 3,\ y = 12x + 18} \]

If forced to select from available, without guaranteed accurate match. Adjust queries or checks might be required to ensure completeness in match with selections seen.

its actually (13 1/2) for the second number in y_b

Thank you for the clarification regarding the value for \( y_b \). Let's reassess the values you provided in the context of this new information.

Given Table of Values Revisited:

| x | y_a | y_b | |---|-----|-----| | 0 | 3 | 18 | | 1 | 6 | 13.5 | | 2 | 9 | 9 |

Analysis of \( y_a \) and \( y_b \):

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

From this, we can calculate the slope:

  • The increase is steady: \( 6 - 3 = 3 \) and \( 9 - 6 = 3 \).

Thus, the equation for \( y_a \) is: \[ y_a = 3x + 3 \]

  1. For \( y_b \):
    • At \( x = 0 \): \( y_b = 18 \)
    • At \( x = 1 \): \( y_b = 13.5 \)
    • At \( x = 2 \): \( y_b = 9 \)

Calculating the differences:

  • From \( x = 0 \) to \( x = 1 \): \( 13.5 - 18 = -4.5 \)
  • From \( x = 1 \) to \( x = 2 \): \( 9 - 13.5 = -4.5 \)

The changes are consistent with a linear equation. We can find the slope: \[ \text{Slope (m)} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 18}{2 - 0} = -4.5 \]

Using the point slope form or y-intercept: Starting with \( y_b = mx + b \): At \( x = 0 \), \( y_b = 18 \) so \( b = 18 \).

Thus, the equation for \( y_b \) would be: \[ y_b = -4.5x + 18 \]

Now we can adjust accordingly with the calculated values. With the values deduced:

Resulting Equations:

We have derived: \[ \begin{align*} y_a &= 3x + 3 \ y_b &= -4.5x + 18 \end{align*} \]

Conclusion and Option Selection:

Now we can see that these equations represent a system of linear relations that correlates with what you provided earlier regarding constant coefficients and base values.

Let's re-examine the available options considering \( y_b = -4.5x + 18 \). Since \( y_b \) wasn't accurately matching previous text options given. Please ensure the respective options reflect equations resembling that derived for clarity.

If you can provide those options again, I can match them more precisely. If not, please consider the derived equations provided above as the correct matching with the newly verified values.