The problem provides a table of $x$ and $y$ values representing a linear relation. We are asked to determine the slope of the linear relation and then find the output value (i.e., the $y$-value) when the input value $x$ is 12.

1. Problem Description

The problem provides a table of $x$ and $y$ values representing a linear relation. We are asked to determine the slope of the linear relation and then find the output value (i.e., the $y$-value) when the input value $x$ is

1

2.

2. Solution Steps

First, we need to find the slope of the linear relation. The slope $m$ can be calculated using any two points $(x_1, y_1)$ and $(x_2, y_2)$ from the table with the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's use the points $(1, 4)$ and $(3, -2)$. Then $x_1 = 1$, $y_1 = 4$, $x_2 = 3$, and $y_2 = -2$. Plugging these values into the slope formula:

$m = \frac{-2 - 4}{3 - 1} = \frac{-6}{2} = -3$

So the slope of the linear relation is -

3.

Now, we need to find the equation of the line in slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. We already have the slope, $m = -3$. We can use one of the points from the table, say $(1, 4)$, to find the y-intercept $b$.

$4 = (-3)(1) + b$

$4 = -3 + b$

$b = 4 + 3 = 7$

So the equation of the line is $y = -3x + 7$.

Finally, we want to determine the output value (y) when the input value is

1

2. We substitute $x = 12$ into the equation:

$y = -3(12) + 7 = -36 + 7 = -29$

3. Final Answer

The slope of the linear relation is -

3. When inputting a value of 12, the output value will be -29.