Philip made tables of values to solve a system of equations. First, he found that the [tex]\( x \)[/tex]-value of the solution was between 0 and 1, and then he found that it was between 0.5 and 1. Next, he made this table.

[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]

Which ordered pair is the best approximation of the exact solution?

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]



Answer :

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]