To determine the equation of the line that best fits the given data points, we can use the linear equation form [tex]\( y = ax + b \)[/tex], where [tex]\( a \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept. Here is a detailed, step-by-step solution:
1. Determine the slope [tex]\( a \)[/tex]:
- The slope [tex]\( a \)[/tex] can be calculated using any two points from the given data set. We'll use the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 3)\)[/tex].
- The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
a = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Plugging in the values from the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 3)\)[/tex]:
[tex]\[
a = \frac{3 - 4}{1 - 0} = \frac{-1}{1} = -1
\][/tex]
2. Determine the y-intercept [tex]\( b \)[/tex]:
- Substitute the slope [tex]\( a = -1 \)[/tex] back into the linear equation [tex]\( y = ax + b \)[/tex] using the point [tex]\((0, 4)\)[/tex].
- Since the point [tex]\((0, 4)\)[/tex] gives us the value of [tex]\( b \)[/tex] directly (as [tex]\( x = 0 \)[/tex]), we have:
[tex]\[
y = -1 \cdot 0 + b = 4 \implies b = 4
\][/tex]
3. Write the final equation:
- Now that we have both the slope [tex]\( a = -1 \)[/tex] and the y-intercept [tex]\( b = 4 \)[/tex], the equation of the line is:
[tex]\[
y = -1x + 4
\][/tex]
- In simplified form, this is:
[tex]\[
y = -x + 4
\][/tex]
Thus, the complete equation describing how the variable [tex]\( y \)[/tex] depends on [tex]\( x \)[/tex] is:
[tex]\[
y = -1x + 4
\][/tex]
So, the final answer to fill in the blanks is:
[tex]\[
y = -1 \cdot x + 4
\][/tex]
or more concisely,
[tex]\[
y = -x + 4
\][/tex]
