Question

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

Table of Values

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

Answer 1

To determine which system of linear equations the table of values represents, we can analyze the data in the table. The table shows two sets of values for \( y_a \) and \( y_b \) corresponding to values of \( x \).

To find the equations for both \( y_a \) and \( y_b \), we can identify the pattern in both columns.

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

It looks like \( y_a \) is increasing by 3 for each increase in \( x \). This is a linear relationship, and we can describe it with the equation:
\[ y_a = 3x + 3 \]

To confirm:
- For \( x = 0 \): \( y_a = 3(0) + 3 = 3 \) ✅
- For \( x = 1 \): \( y_a = 3(1) + 3 = 6 \) ✅
- For \( x = 2 \): \( y_a = 3(2) + 3 = 9 \) ✅

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

Here, \( y_b \) is decreasing as \( x \) increases. The changes in \( y_b \) are:
- From \( 18 \) to \( 13 \): decrease of 5
- From \( 13 \) to \( 9 \): decrease of 4

To find a linear equation, we can use the first and last points:
Using points \( (0, 18) \) and \( (2, 9) \), the slope \( m \) can be calculated as:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 18}{2 - 0} = \frac{-9}{2} = -\frac{9}{2}
\]

Thus, the equation of the line can be expressed in point-slope form:
Using point \( (0, 18) \):
\[
y_b - 18 = -\frac{9}{2}(x - 0) \Rightarrow y_b = -\frac{9}{2}x + 18
\]

But we need an integer slope. Therefore, for simplicity:
\[
y_b = -4.5x + 18
\]

Depending on how the equations are simplified in the possible responses, we can express it, using integer coefficients as:
\[
y_b = -\frac{9}{2}x + 18 \text{ or } 9x + y_b = 36
\]

Now let's match the original options:

1. **Option** \( \{y = 3x + 3, y = 4x + 12\} \) (not correct)
2. **Option** \( \{y = 3x + 3, y = 4x - 18\} \) (not correct)
3. **Option** \( \{y = -3x + 3, y = 4x + 12\} \) (not correct)

None of the options provided match our equations directly, indicating there may have been a mistake in obtaining the answers from the choices provided.

If the correct equations don't perfectly match any, the answer might be:
\{ y=3x+3 , y=4x-12 \} if it were matched correctly based on original numerical transformations and minimizing errors.

If necessary, please clarify table values if they differ or re-verify data points.