Sure, let's determine the equation of the line that passes through the points (5, -5) and (-8, 8) step-by-step.
### Step 1: Find the Slope (m)
The slope 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]
For the points (5, -5) and (-8, 8):
[tex]\[ x_1 = 5, \ y_1 = -5, \ x_2 = -8, \ y_2 = 8 \][/tex]
Substituting these values:
[tex]\[ m = \frac{8 - (-5)}{-8 - 5} \][/tex]
[tex]\[ m = \frac{8 + 5}{-8 - 5} \][/tex]
[tex]\[ m = \frac{13}{-13} \][/tex]
[tex]\[ m = -1 \][/tex]
So, the slope of the line is [tex]\( m = -1 \)[/tex].
### Step 2: Determine the Y-Intercept (b)
The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
We already have [tex]\( m \)[/tex]. To find [tex]\( b \)[/tex], we use one of the points. Let's use (5, -5).
Substitute [tex]\( x = 5 \)[/tex], [tex]\( y = -5 \)[/tex], and [tex]\( m = -1 \)[/tex] into the equation [tex]\( y = mx + b \)[/tex]:
[tex]\[ -5 = (-1)(5) + b \][/tex]
[tex]\[ -5 = -5 + b \][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[ -5 + 5 = b \][/tex]
[tex]\[ b = 0 \][/tex]
So, the y-intercept [tex]\( b \)[/tex] is 0.
### Step 3: Write the Equation
With [tex]\( m = -1 \)[/tex] and [tex]\( b = 0 \)[/tex], the equation of the line is:
[tex]\[ y = -x \][/tex]
Therefore, the equation of the line that passes through the points (5, -5) and (-8, 8) is:
[tex]\[ y = -x \][/tex]
