To determine the equation of the line with a slope of 4 that contains the point [tex]\((5, 8)\)[/tex], we can use the point-slope form of the equation of a line, which is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the given point.
Here, the slope [tex]\( m \)[/tex] is 4, and the given point [tex]\((x_1, y_1)\)[/tex] is [tex]\((5, 8)\)[/tex]. Plugging these values into the point-slope formula, we get:
[tex]\[ y - 8 = 4(x - 5) \][/tex]
Next, we need to simplify this equation to the slope-intercept form [tex]\( y = mx + b \)[/tex]. Let’s distribute the slope (4) on the right-hand side and then solve for [tex]\( y \)[/tex].
First, distribute the 4:
[tex]\[ y - 8 = 4x - 20 \][/tex]
Next, isolate [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 20 + 8 \][/tex]
[tex]\[ y = 4x - 12 \][/tex]
So, the equation of the line in slope-intercept form is [tex]\( y = 4x - 12 \)[/tex].
Now, we need to select the correct option from the given choices:
a. [tex]\( y = 4x - 27 \)[/tex]
b. [tex]\( y = 4x - 12 \)[/tex]
c. [tex]\( y = 4x + 28 \)[/tex]
d. [tex]\( y = 4x + 8 \)[/tex]
The correct answer is clearly:
b. [tex]\( y = 4x - 12 \)[/tex]
