Let's analyze how Samuel wrote the equation of the line in slope-intercept form using the given points [tex]\((-7, 13)\)[/tex], [tex]\((-5, 5)\)[/tex], [tex]\((-3, -3)\)[/tex], and [tex]\((-1, -11)\)[/tex]:
Step 1: Calculate the Slope [tex]\( m \)[/tex]
To find the slope of the line, Samuel used two points, [tex]\((-5, 5)\)[/tex] and [tex]\((-3, -3)\)[/tex].
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points:
[tex]\[ m = \frac{-3 - 5}{-3 - (-5)} = \frac{-8}{2} = -4 \][/tex]
Thus, the slope [tex]\( m \)[/tex] is [tex]\( -4 \)[/tex].
Step 2: Calculate the y-intercept [tex]\( b \)[/tex]
Now that we have the slope, we need to find the y-intercept [tex]\( b \)[/tex]. We use one of the points on the line, for instance, [tex]\((-5, 5)\)[/tex].
The slope-intercept form of a line is:
[tex]\[ y = mx + b \][/tex]
Substitute [tex]\( m = -4 \)[/tex], [tex]\( x = -5 \)[/tex], and [tex]\( y = 5 \)[/tex] into the equation and solve for [tex]\( b \)[/tex]:
[tex]\[ 5 = -4(-5) + b \][/tex]
[tex]\[ 5 = 20 + b \][/tex]
[tex]\[ b = 5 - 20 \][/tex]
[tex]\[ b = -15 \][/tex]
Thus, the y-intercept [tex]\( b \)[/tex] is [tex]\( -15 \)[/tex].
Step 3: Write the equation in slope-intercept form
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 in the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = -4x - 15 \][/tex]
So, the equation of the line is:
[tex]\[ y = -4x - 15 \][/tex]
There was a slight typo in Samuel's final equation; it should be [tex]\( y = -4x - 15 \)[/tex] instead of [tex]\( y = -15x - 4 \)[/tex].
In summary, we determined the slope to be [tex]\( -4 \)[/tex], the y-intercept [tex]\( b \)[/tex] to be [tex]\( -15 \)[/tex], and combined them to get the equation [tex]\( y = -4x - 15 \)[/tex].
