Question

Writing Simultaneous Linear Equations Quick Check
1 of 51 of 5 Items
Question
Use the table to answer the question.

Table of Values

Values of Function
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

Answer 1

To determine the system of linear equations represented by the given table, we need to identify the equations corresponding to \(y_a\) and \(y_b\) based on the values for \(x\).

Let's start with the values provided for \(y_a\):

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

These values suggest a linear relationship. To find the equation for \(y_a\):

1. We can see that as \(x\) increases by 1, \(y_a\) increases by 3. This indicates that the slope (m) is 3.
2. The formula for a linear equation is \(y = mx + b\). We can use one of the points to find \(b\).

Using the point (0, 3):
\[
3 = 3(0) + b \Rightarrow b = 3
\]

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

Now let's look at \(y_b\):

- When \(x = 0\), \(y_b = 18\)
- When \(x = 1\), \(y_b = 13\)
- When \(x = 2\), \(y_b = 9\)

The change in \(y_b\) when \(x\) increases by 1 is a decrease of 5 (from 18 to 13, and then from 13 to 9). Therefore, the slope \(m\) for \(y_b\) is -5.

We can also find the y-intercept \(b\) using the point (0, 18):
\[
18 = -5(0) + b \Rightarrow b = 18
\]

So, the equation for \(y_b\) is:
\[
y_b = -5x + 18
\]

Putting it all together, we can write the system of equations as:
\[
\begin{cases}
y = 3x + 3 \\
y = -5x + 18
\end{cases}
\]

From the list of responses you provided, the first response \( y = 3x + 3 \) matches, while the second equation \( y = -4x + 12 \) seems to differ from what we derived.

The closest matching system among your options appears to be:
- \(\{y = 3x + 3, y = -4x + 12\}\)

However, since we determined the second equation doesn't match what we have. Please check the options again, or the values in the table as there might be a small error in the listed options. The correct equations based on the values provided appear to be:
- \(y = 3x + 3\)
- \(y = -5x + 18\) (not listed).

If we need to choose based on what’s available, confirm the equations or their representation and use what best matches.