To determine how [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are related, follow these steps:
1. Understand the Form of the Relationship:
We are looking for a linear relationship of the form:
[tex]\[
y = mx + c
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
2. Calculate the Slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] can be determined by observing the change in [tex]\( y \)[/tex] with respect to the change in [tex]\( x \)[/tex]. We can use any two points to calculate this.
Let’s use the points [tex]\( (-5, 17) \)[/tex] and [tex]\( (-4, 14) \)[/tex]:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{14 - 17}{-4 - (-5)} = \frac{-3}{1} = -3
\][/tex]
Thus, the slope [tex]\( m \)[/tex] is [tex]\(-3\)[/tex].
3. Determine the y-Intercept [tex]\( c \)[/tex]:
Use one of the given points and the slope to solve for [tex]\( c \)[/tex]. Let’s use the point [tex]\((-5, 17)\)[/tex]:
[tex]\[
y = mx + c \implies 17 = (-3)(-5) + c
\][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[
17 = 15 + c \implies c = 17 - 15 = 2
\][/tex]
Thus, the y-intercept [tex]\( c \)[/tex] is [tex]\( 2 \)[/tex].
4. Formulate the Equation:
Using the obtained values of [tex]\( m \)[/tex] and [tex]\( c \)[/tex], we can write the equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
y = -3x + 2
\][/tex]
Therefore, the completed equation is:
[tex]\[
y = -3x + 2
\][/tex]
