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?

{y=2xy=3x+1/2

{y=2x+3y=1/2x

{y=2xy=1/2x+3

{y=12xy=2x+3

Answer 1

To determine which system of linear equations corresponds to the given table of values, we need to analyze the points provided for both \(y_a\) and \(y_b\):

\[
\begin{array}{c|c|c}
x & y_a & y_b \\
\hline
0 & 0 & 3 \\
1 & 2 & 3.5 \\
2 & 4 & 4 \\
\end{array}
\]

### Step 1: Check \(y_a\)

First, let's find the equation that matches \(y_a\):
- At \(x = 0\), \(y_a = 0\)
- At \(x = 1\), \(y_a = 2\)
- At \(x = 2\), \(y_a = 4\)

The pattern suggests a relationship \(y_a = 2x\). We can verify:
- For \(x = 0\): \(y_a = 2(0) = 0\)
- For \(x = 1\): \(y_a = 2(1) = 2\)
- For \(x = 2\): \(y_a = 2(2) = 4\)

Thus, we have \(y_a = 2x\).

### Step 2: Check \(y_b\)

Now, let's derive the equation for \(y_b\):
- At \(x = 0\), \(y_b = 3\)
- At \(x = 1\), \(y_b = 3.5\)
- At \(x = 2\), \(y_b = 4\)

We can observe that \(y_b\) seems to increase at a decreasing rate. To derive the equation, let's calculate the differences:

From \(x = 0\) to \(x = 1\): \(3.5 - 3 = 0.5\)
From \(x = 1\) to \(x = 2\): \(4 - 3.5 = 0.5\)

This difference is constant, suggesting a linear relationship. To find the slope:
\[
\text{slope} = \frac{4 - 3}{2 - 0} = \frac{1}{2}
\]

Thus, the equation can be expressed in the slope-intercept form:
\[
y_b = \frac{1}{2}x + 3
\]

### Step 3: Set of Equations

Now, we can summarize our findings:
- \(y_a = 2x\)
- \(y_b = \frac{1}{2}x + 3\)

The corresponding system of equations is:

\[
\begin{cases}
y = 2x \\
y = \frac{1}{2}x + 3
\end{cases}
\]

### Step 4: Match with Provided Options

Now, let's check the options provided:
- \( \{y = 2x, y = \frac{1}{2}x + 3\} \)

This matches with the third option:
**{y=2x, y=1/2x+3}**

Thus, the correct choice from the options given is:
\[
\{y = 2x, y = \frac{1}{2}x + 3\}
\]