To determine the slope of the line that passes through the points [tex]\((-4, -4)\)[/tex] and [tex]\( (12, 16)\)[/tex], we need to use the formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
First, we identify the coordinates:
- [tex]\((x_1, y_1) = (-4, -4)\)[/tex]
- [tex]\((x_2, y_2) = (12, 16)\)[/tex]
Next, we calculate the difference in the y-coordinates ([tex]\(\Delta y\)[/tex]) and the difference in the x-coordinates ([tex]\(\Delta x\)[/tex]):
[tex]\[
\Delta y = y_2 - y_1 = 16 - (-4) = 16 + 4 = 20
\][/tex]
[tex]\[
\Delta x = x_2 - x_1 = 12 - (-4) = 12 + 4 = 16
\][/tex]
Now we substitute these differences into the slope formula:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{20}{16}
\][/tex]
To simplify the fraction [tex]\(\frac{20}{16}\)[/tex], we divide both the numerator and the denominator by their greatest common divisor, which is 4:
[tex]\[
\frac{20 \div 4}{16 \div 4} = \frac{5}{4}
\][/tex]
Therefore, the slope of the line that passes through the points [tex]\((-4, -4)\)[/tex] and [tex]\( (12, 16) \)[/tex] is:
[tex]\[
\boxed{\frac{5}{4}}
\][/tex]
