Answered

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

\begin{tabular}{|c|c|c|}
\hline[tex]$x$[/tex] & [tex]$y=2x-3$[/tex] & [tex]$y=-5x+2$[/tex] \\
\hline 0.5 & -2 & -0.5 \\
\hline 0.6 & -1.8 & -1 \\
\hline 0.7 & -1.6 & -1.5 \\
\hline 0.8 & -1.4 & -2 \\
\hline 0.9 & -1.2 & -2.5 \\
\hline 1.0 & -1 & -3 \\
\hline
\end{tabular}

Which ordered pair is the best approximation of the solution?

A. [tex]$(0.9, -2.1)$[/tex]
B. [tex]$(0.5, -1.8)$[/tex]
C. [tex]$(0.7, -1.5)$[/tex]
D. [tex]$(0.6, -1.4)$[/tex]



Answer :

GinnyAnswer
To determine the best approximation for the solution to the system of equations using the table provided, we need to find the [tex]\(x\)[/tex]-value in the table where the [tex]\(y\)[/tex]-values of both equations are closest to each other.

Let's analyze the table:
[tex]\[ \begin{array}{|c|c|c|} \hline x & y=2x-3 & y=-5x+2 \\ \hline 0.5 & -2 & -0.5 \\ 0.6 & -1.8 & -1 \\ 0.7 & -1.6 & -1.5 \\ 0.8 & -1.4 & -2 \\ 0.9 & -1.2 & -2.5 \\ 1.0 & -1 & -3 \\ \hline \end{array} \][/tex]

We compare the pairs of [tex]\(y\)[/tex]-values corresponding to each [tex]\(x\)[/tex] value:

- For [tex]\(x = 0.5\)[/tex]:
[tex]\[ y = 2(0.5)-3 = -2 \\ y = -5(0.5)+2 = -0.5 \\ \text{Difference} = | -2 - (-0.5)| = 1.5 \][/tex]

- For [tex]\(x = 0.6\)[/tex]:
[tex]\[ y = 2(0.6)-3 = -1.8 \\ y = -5(0.6)+2 = -1 \\ \text{Difference} = | -1.8 - (-1)| = 0.8 \][/tex]

- For [tex]\(x = 0.7\)[/tex]:
[tex]\[ y = 2(0.7)-3 = -1.6 \\ y = -5(0.7)+2 = -1.5 \\ \text{Difference} = | -1.6 - (-1.5)| = 0.1 \][/tex]

- For [tex]\(x = 0.8\)[/tex]:
[tex]\[ y = 2(0.8)-3 = -1.4 \\ y = -5(0.8)+2 = -2 \\ \text{Difference} = | -1.4 - (-2)| = 0.6 \][/tex]

- For [tex]\(x = 0.9\)[/tex]:
[tex]\[ y = 2(0.9)-3 = -1.2 \\ y = -5(0.9)+2 = -2.5 \\ \text{Difference} = | -1.2 - (-2.5)| = 1.3 \][/tex]

- For [tex]\(x = 1.0\)[/tex]:
[tex]\[ y = 2(1)-3 = -1 \\ y = -5(1)+2 = -3 \\ \text{Difference} = | -1 - (-3)| = 2 \][/tex]

The closest [tex]\(y\)[/tex]-values occur when [tex]\(x = 0.7\)[/tex] with [tex]\(y\)[/tex]-values [tex]\(-1.6\)[/tex] and [tex]\(-1.5\)[/tex] giving a difference of [tex]\(0.1\)[/tex].

To approximate the corresponding ordered pair for [tex]\(x = 0.7\)[/tex]:

The approximate [tex]\(y\)[/tex]-value is around the middle of [tex]\(-1.6\)[/tex] and [tex]\(-1.5\)[/tex]:

[tex]\[ y \approx \left(\frac{-1.6 + (-1.5)}{2}\right) = \left(\frac{-3.1}{2}\right) = -1.55 \][/tex]

Thus, the best approximation of the solution in the table is:

[tex]\[ (0.7, -1.55) \][/tex]

Among the given answer choices, the closest one is:

C. [tex]\((0.7, -1.5)\)[/tex]

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