To find the equation of the line passing through the given points [tex]\((7, 8)\)[/tex] and [tex]\((0, -8)\)[/tex], we need to follow a step-by-step approach:
### Step 1: Calculate the Slope
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the given points [tex]\((7, 8)\)[/tex] and [tex]\((0, -8)\)[/tex]:
[tex]\[ m = \frac{-8 - 8}{0 - 7} \][/tex]
[tex]\[ m = \frac{-16}{-7} \][/tex]
[tex]\[ m = \frac{16}{7} \][/tex]
### Step 2: Use the Point-Slope Formula
The point-slope form of a line equation is given by:
[tex]\[ y - y_1 = m (x - x_1) \][/tex]
We can use either of the provided points [tex]\((7, 8)\)[/tex] or [tex]\((0, -8)\)[/tex]. Let's use the point [tex]\((7, 8)\)[/tex]:
[tex]\[ y - 8 = \frac{16}{7}(x - 7) \][/tex]
### Step 3: Simplify the Equation
Expand and simplify the right-hand side:
[tex]\[ y - 8 = \frac{16}{7}x - \frac{16}{7} \cdot 7 \][/tex]
[tex]\[ y - 8 = \frac{16}{7}x - \frac{112}{7} \][/tex]
[tex]\[ y - 8 = \frac{16}{7}x - 16 \][/tex]
Finally, add 8 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{16}{7}x - 16 + 8 \][/tex]
[tex]\[ y = \frac{16}{7}x - 8 \][/tex]
Therefore, the equation of the line passing through the points [tex]\((7, 8)\)[/tex] and [tex]\((0, -8)\)[/tex] in the simplified slope-intercept form is:
[tex]\[ \boxed{y = \frac{16}{7}x - 8} \][/tex]
