Certainly! Let's determine the linear equation that relates [tex]\( x \)[/tex] and [tex]\( y \)[/tex] based on the given table of values.
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -1 \\
\hline
2 & -2 \\
\hline
3 & -3 \\
\hline
4 & -4 \\
\hline
\end{array}
\][/tex]
To find the linear equation [tex]\( y = mx + b \)[/tex], we need to calculate the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex].
### Finding the Slope ([tex]\( m \)[/tex])
The slope [tex]\( m \)[/tex] can be determined using any two points from the table. Let's use the points [tex]\((1, -1)\)[/tex] and [tex]\((2, -2)\)[/tex].
The formula to calculate the slope ([tex]\( m \)[/tex]) is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the coordinates of the two points:
[tex]\[
m = \frac{-2 - (-1)}{2 - 1} = \frac{-2 + 1}{1} = \frac{-1}{1} = -1
\][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\(-1\)[/tex].
### Finding the Y-Intercept ([tex]\( b \)[/tex])
The y-intercept [tex]\( b \)[/tex] can be found by plugging in the slope [tex]\( m \)[/tex] and the coordinates of one of the points into the equation [tex]\( y = mx + b \)[/tex]. Let's use the point [tex]\((1, -1)\)[/tex].
The equation becomes:
[tex]\[
-1 = (-1)(1) + b
\][/tex]
[tex]\[
-1 = -1 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = -1 + 1 = 0
\][/tex]
So, the y-intercept [tex]\( b \)[/tex] is [tex]\( 0 \)[/tex].
### Forming the Equation
Now that we have the slope and the y-intercept, we can write the linear equation:
[tex]\[
y = -1x + 0
\][/tex]
Simplifying this, we get:
[tex]\[
y = -x
\][/tex]
Thus, the linear equation that represents the relationship in the table is:
[tex]\[
y = -x
\][/tex]
