Richard 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.

\begin{tabular}{|l|l|l|}
\hline
[tex]$x$[/tex] & [tex]$y=-x+5$[/tex] & [tex]$y=6x-1$[/tex] \\
\hline
0.5 & 4.5 & 2 \\
\hline
0.6 & 4.4 & 2.6 \\
\hline
0.7 & 4.3 & 3.2 \\
\hline
0.8 & 4.2 & 3.8 \\
\hline
0.9 & 4.1 & 4.4 \\
\hline
1.0 & 4 & 5 \\
\hline
\end{tabular}

Which ordered pair is the best approximation of the solution?



Answer :

To find the best approximation of the solution, we first need to determine the ordered pair [tex]\((x, y)\)[/tex] where the values of [tex]\(y\)[/tex] from the two functions [tex]\(-x + 5\)[/tex] and [tex]\(6x - 1\)[/tex] are closest. We start by organizing our data and then finding the differences between the [tex]\(y\)[/tex]-values for various [tex]\(x\)[/tex]-values.

Step 1: Set up the table from the given data.

| [tex]\(x\)[/tex] | [tex]\(y = -x + 5\)[/tex] | [tex]\(y = 6x - 1\)[/tex] |
|---|---|---|
| 0.5 | 4.5 | 2 |
| 0.6 | 4.4 | 2.6 |
| 0.7 | 4.3 | 3.2 |
| 0.8 | 4.2 | 3.8 |
| 0.9 | 4.1 | 4.4 |
| 1.0 | 4 | 5 |

Step 2: Calculate the absolute differences [tex]\( | y_1 - y_2 | \)[/tex] for each pair [tex]\((x, y_1, y_2)\)[/tex].

[tex]\[ \begin{align*} |4.5 - 2| & = 2.5 \\ |4.4 - 2.6| & = 1.8 \\ |4.3 - 3.2| & = 1.1 \\ |4.2 - 3.8| & = 0.4 \\ |4.1 - 4.4| & = 0.3 \\ |4 - 5| & = 1 \\ \end{align*} \][/tex]

Step 3: Identify the smallest difference.

The smallest difference calculated is [tex]\(0.3\)[/tex]. This difference occurs for the [tex]\(x\)[/tex]-value [tex]\(0.9\)[/tex].

Step 4: Determine the ordered pair corresponding to the smallest difference.

At [tex]\(x = 0.9\)[/tex], the value of [tex]\(y\)[/tex] calculated from [tex]\(y = -x + 5\)[/tex] is [tex]\(4.1\)[/tex]. Therefore, the ordered pair is [tex]\((0.9, 4.1)\)[/tex].

Conclusion:

The best approximation of the solution to the system of equations [tex]\(y = -x + 5\)[/tex] and [tex]\(y = 6x - 1\)[/tex] within the given range is [tex]\((0.9, 4.1)\)[/tex].