To determine the equation that describes how [tex]\(X\)[/tex] and [tex]\(y\)[/tex] are related, we will use the given set of points to find a linear relationship of the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
### Step 1: Calculate the Slope (m)
We can determine the slope ([tex]\(m\)[/tex]) using any two points from the given data. For consistency, let's use the points [tex]\((-3, -25)\)[/tex] and [tex]\((-2, -18)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the values of the points into the slope formula:
[tex]\[
m = \frac{-18 - (-25)}{-2 - (-3)} = \frac{-18 + 25}{-2 + 3} = \frac{7}{1} = 7
\][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(7\)[/tex].
### Step 2: Determine the Y-Intercept (b)
We can use the slope we just found ([tex]\(m = 7\)[/tex]) and one of the points to solve for the y-intercept ([tex]\(b\)[/tex]). Let's use the point [tex]\((0, -4)\)[/tex], which is the y-intercept because [tex]\(x = 0\)[/tex]:
[tex]\[
y = mx + b \implies -4 = 7(0) + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -4
\][/tex]
### Step 3: Form the Equation
Now that we have both the slope ([tex]\(m\)[/tex]) and the y-intercept ([tex]\(b\)[/tex]), we can write the equation of the line:
[tex]\[
y = 7x - 4
\][/tex]
### Verification with Other Points
For completeness, let's verify this equation with another point from the table, say [tex]\((1, 3)\)[/tex]:
Substitute [tex]\(x = 1\)[/tex] into the equation [tex]\(y = 7x - 4\)[/tex]:
[tex]\[
y = 7(1) - 4 = 7 - 4 = 3
\][/tex]
The calculation matches the given point [tex]\((1, 3)\)[/tex].
Thus, the equation describing how [tex]\(X\)[/tex] and [tex]\(y\)[/tex] are related is:
[tex]\[
\boxed{y = 7x - 4}
\][/tex]
