The problem asks us to analyze a table of input values $x$ and output values $y$ to determine the rule that relates them. We need to express this rule in words and as an algebraic expression. Then, we need to use the rule to complete the table, specifically finding the values of $x$ when $y = 59$ and $y$ when $x = 13$.

1. Problem Description

The problem asks us to analyze a table of input values $x$ and output values $y$ to determine the rule that relates them. We need to express this rule in words and as an algebraic expression. Then, we need to use the rule to complete the table, specifically finding the values of $x$ when $y = 59$ and $y$ when $x = 13$.

2. Solution Steps

a. Write the rule in words.

Looking at the first few values, we see that when $x=1$, $y=1$; when $x=2$, $y=4$; and when $x=3$, $y=7$. The output $y$ increases by 3 for each increase of 1 in the input $x$. We can also see that $y = 3x - 2$.

The rule in words is: Multiply the input by 3 and subtract 2 to get the output.

b. Write the rule in algebraic expressions.

The algebraic expression for the rule is $y = 3x - 2$.

c. Use the rule to complete the table.

When $x = 8$, $y = 3(8) - 2 = 24 - 2 = 22$.

When $x = 13$, $y = 3(13) - 2 = 39 - 2 = 37$.

When $y = 59$, $59 = 3x - 2$. Adding 2 to both sides gives $61 = 3x$, so $x = 61/3 \approx 20.33$. However, since the other x values are integers, we might assume that the x and y values should be integers. With an integer x value, y = 59 is not possible based on the formula. So, we reexamine the table. The difference between each y value is 3, and since we have the y value when x is 13, we can use the equation $y = 3x - 2$ to find the x value when y =

5

9. The x values are all integers, so we will use the correct rule for $y=59$ to find $x$.

$y = 3x - 2$

$59 = 3x - 2$

$61 = 3x$

$x = 61/3$

The table might have an error or typo in it because $x$ has to be an integer for $y=59$. Let's assume the equation $y = 3x - 2$ is correct and the $x$ values should be integers. Let's find the missing table value assuming the y value at x = 8 is an error. The table should be:

Term/input (x) 1 2 3 4 5 6 7 8 13 x

Result/output (y) 1 4 7 10 13 16 19 22 y 59

When x = 8, y = 3(8) - 2 = 24 - 2 = 22

When x = 13, y = 3(13) - 2 = 39 - 2 = 37

If y = 59, then

59 = 3x - 2

61 = 3x

x = 61/3 = 20.33

Since x needs to be an integer, we will say x =

2

1. If x = 21, y = 3(21) - 2 = 63 - 2 =

6

1. Based on the initial table, it is more likely the y value at $x = 13$ has an error. We will keep using the rule $y = 3x - 2$

When $x = 13$, $y = 3(13) - 2 = 39 - 2 = 37$

d. Find the value of x and y in the table.

$x = 61/3$ when $y = 59$. If we have to round to an integer then $x \approx 20$

$y = 37$ when $x = 13$.

3. Final Answer

a. The rule in words: Multiply the input by 3 and subtract 2 to get the output.

b. The rule in algebraic expressions: $y = 3x - 2$

c. Complete the table: When $x = 13$, $y = 37$.

d. The values of $x$ and $y$ in the table: $x = \frac{61}{3}$ (or approximately 20 if we round to the nearest integer) and $y = 37$.