To find the ordered pair that best approximates the exact solution to the system of equations, we follow these steps:
1. Examine the given table of values:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
$x$ & $y=-2x+3$ & $y=5x-1$ \\
\hline
0.5 & 2 & 1.5 \\
\hline
0.6 & 1.8 & 2 \\
\hline
0.7 & 1.6 & 2.5 \\
\hline
0.8 & 1.4 & 3 \\
\hline
0.9 & 1.2 & 3.5 \\
\hline
1.0 & 1 & 4 \\
\hline
\end{tabular}
\][/tex]
2. Identify the point where the [tex]$y$[/tex]-values of the two equations are the closest. This requires comparing the absolute differences between the [tex]$y$[/tex]-values for all [tex]$x$[/tex]-values in the table.
[tex]\[
\begin{aligned}
&\text{For } x = 0.5, \ |2 - 1.5| = 0.5 \\
&\text{For } x = 0.6, \ |1.8 - 2| = 0.2 \\
&\text{For } x = 0.7, \ |1.6 - 2.5| = 0.9 \\
&\text{For } x = 0.8, \ |1.4 - 3| = 1.6 \\
&\text{For } x = 0.9, \ |1.2 - 3.5| = 2.3 \\
&\text{For } x = 1.0, \ |1 - 4| = 3 \\
\end{aligned}
\][/tex]
3. From these calculations, the smallest absolute difference between the [tex]$y$[/tex]-values occurs when [tex]$x = 0.6$[/tex], with a difference of [tex]$0.2$[/tex]. The corresponding [tex]$y$[/tex]-values are [tex]$1.8$[/tex] and [tex]$2$[/tex].
4. Compare the choices provided:
- A. [tex]$(0.9, 1.2)$[/tex]
- B. [tex]$(0.5, 1.2)$[/tex]
- C. [tex]$(0.8, 2.3)$[/tex]
- D. [tex]$(0.6, 1.9)$[/tex]
From these options, the closest match found in the examination is when [tex]$x = 0.6$[/tex] with [tex]$y$[/tex]-values [tex]$1.8$[/tex] and [tex]$2$[/tex].
Thus, the best approximation of the exact solution is:
[tex]\[ \boxed{(0.6, 1.9)} \][/tex]
