Questions: Super equations. First she found that the (x)-value of the solution was between 0 and 1, and then she found that it was between 0 and 0.5. Next, she made this table. (x) (y=-3 x+4) (y=5 x+1) 0 4 1 0.1 3.7 1.5 0.2 3.4 2 0.3 3.1 2.5 0.4 2.8 3 0.5 2.5 3.5 Which ordered pair is the best approximation of the exact solution? A. ((0.4,3.7)) B. ((0.1,2.1)) C. ((0.2,3.4)) D. ((0.4,2.9))

Super equations. First she found that the (x)-value of the solution was between 0 and 1, and then she found that it was between 0 and 0.5. Next, she made this table.

(x) (y=-3 x+4) (y=5 x+1)

0 4 1

0.1 3.7 1.5

0.2 3.4 2

0.3 3.1 2.5

0.4 2.8 3

0.5 2.5 3.5

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

A. ((0.4,3.7))

B. ((0.1,2.1))

C. ((0.2,3.4))

D. ((0.4,2.9))

Transcript text: Super equations. First she found that the $x$-value of the solution was between 0 and 1, and then she found that it was between 0 and 0.5. Next, she made this table. \begin{tabular}{|c|c|c|} \hline$x$ & $y=-3 x+4$ & $y=5 x+1$ \\ \hline 0 & 4 & 1 \\ \hline 0.1 & 3.7 & 1.5 \\ \hline 0.2 & 3.4 & 2 \\ \hline 0.3 & 3.1 & 2.5 \\ \hline 0.4 & 2.8 & 3 \\ \hline 0.5 & 2.5 & 3.5 \\ \hline \end{tabular} Which ordered pair is the best approximation of the exact solution? A. $(0.4,3.7)$ B. $(0.1,2.1)$ C. $(0.2,3.4)$ D. $(0.4,2.9)$

Solution

Solution Steps

To find the best approximation of the exact solution, we need to identify the point where the two equations $ y = -3x + 4 $ and $ y = 5x + 1 $ intersect. This means finding the $ x $ value where both equations yield the same $ y $ value. We can do this by iterating through the given table and checking for the closest match between the $ y $ values of the two equations.

Step 1: Define the Equations

We are given two equations: \[ y = -3x + 4 \] \[ y = 5x + 1 \]

Step 2: Create a Table of Values

We have a table of values for $ x $ ranging from 0 to 0.5:

\[ \begin{array}{|c|c|c|} \hline x & y = -3x + 4 & y = 5x + 1 \\ \hline 0 & 4 & 1 \\ 0.1 & 3.7 & 1.5 \\ 0.2 & 3.4 & 2 \\ 0.3 & 3.1 & 2.5 \\ 0.4 & 2.8 & 3 \\ 0.5 & 2.5 & 3.5 \\ \hline \end{array} \]

Step 3: Find the Closest Intersection

We need to find the $ x $ value where the difference between $ y = -3x + 4 $ and $ y = 5x + 1 $ is minimized. Calculating the differences:

\[ \begin{array}{|c|c|c|c|} \hline x & y = -3x + 4 & y = 5x + 1 & |y_1 - y_2| \\ \hline 0 & 4 & 1 & 3 \\ 0.1 & 3.7 & 1.5 & 2.2 \\ 0.2 & 3.4 & 2 & 1.4 \\ 0.3 & 3.1 & 2.5 & 0.6 \\ 0.4 & 2.8 & 3 & 0.2 \\ 0.5 & 2.5 & 3.5 & 1 \\ \hline \end{array} \]

The smallest difference is $ 0.2 $ at $ x = 0.4 $.