Final answer:
For each given point and slope, we can determine the equation by substituting the values into the slope-intercept form. The correct equations for the given scenarios are: a) y = -4x - 3, b) y = 12x - 35, c) y = -x + 2, and d) x = 2.
Step-by-step explanation:
To write the equation of a line in slope-intercept form (y = mx + b), we need to know the slope (m) and a point on the line (x, y).
The slope-intercept form allows us to easily identify the slope and y-intercept of the line.
a) The line through (-2, 5) with a slope of -4 can be written as y = -4x + b.
To find the value of b, substitute the coordinates of the given point into the equation.
5 = -4(-2) + b. Solving for b, we get b = -3.
Therefore, the equation is y = -4x - 3.
b) The line through (3, 1) with a slope of 12 can be written as y = 12x + b.
Substitute the coordinates of the point into the equation to find the value of b.
1 = 12(3) + b. Solving for b, we get b = -35.
Therefore, the equation is y = 12x - 35.
c) The line through (3, -1) with a slope of -1 can be written as y = -1x + b.
Substitute the point's coordinates into the equation to find b.
-1 = -1(3) + b. Solving for b, we get b = 2.
Therefore, the equation is y = -x + 2.
d) The line through (2, 5) with an undefined slope is a vertical line passing through x = 2.
Therefore, the equation is x = 2.
Therefore, the correct equations for the given scenarios are: a) y = -4x - 3, b) y = 12x - 35, c) y = -x + 2, and d) x = 2.