Slope intercept form is y = mx +b ; where m = slope of the line and b = y-intercept.
1) By definition, parallel lines have the same slope. The line in question passes through point (-3,5) and is parallel to:
y = -(1/2)x + 4 (in slope-intercept form)
We know that the slope of the line in question is -1/2, so we know the first part of the equation is y = (-1/2)x ; now we need to find out what b is. Well, we have y, and we have x, so plug in the values for them and solve for b in the equation.
y = (-1/2)x + b
5 = (-1/2)(-3) + b
5 = (3/2) + b
5 - (3/2) = b
(10/2) - (3/2) = b
7/2 = b
Equation is : y = (-1/2)x + (7/2)
2) Again, parallel lines always have the same slope. This time, we're given a line that passes through point (-7,3) and is parallel to the line x=4. By definition, a line in the form x = a, will have no y intercept. We have the x value of -7 as a point on the line in question, which means the form of the line is x = -7. Also, the slope of a vertical line is undefined. Think about the formula for slope is: rise/run. A vertical line rises, but has a run of 0. We can't divide by 0, so the slope is undefined, therefore, there is no slope intercept form for the line in question.
3)Perpendicular lines have slopes that are negative reciprocals of each other. Line passing through (5,-1) perpendicular to y = 4x -7
Slope of known line = 4
Slope of unknown line must = -1/4
Like problem 1, we now have the slope, and points x and y, which means we just have to solve for b in the slope intercept formula:
y = mx + b
-1 = -(1/4)(5) + b
-1 = -(5/4) + b
-1 + 5/4 = b
(-4/4) + (5/4) = b
1/4 = b
So slope intercept form is :
y = -(1/4)x + (1/4)
4) Line y = 3 is exactly horizontal, because y will never change. So, if the known line, is exactly horizontal, the unknown line that is exactly perpendicular to the known line will be vertical. Like problem 2, the unknown line has a slope that is undefined. But we can write the equation for the line. If a vertical line passes through point (4, -2), than that line must have a formula in the form of x = a, where a = x coordinate of the point in which the line passes through.
So the equation of the unknown line is :
x = 4
5) Parallel lines have the exact same slope values. So :
So, whatever k is, the slope must equal 3, when the equations are in the same format.
The unknown k is in a slightly different format than slope-intercept form. The equation 2y = kx + 9 can be solved for y:
2y = kx + 9
y = (k/2)x + (9/2)
Now you have a slope of k/2
Set it equal to 3, which it has to equal, and solve for k:
k/2 = 3
k = 3*2
k = 6
Therefore, the only value of k that would make the equation 2y = kx + 9 parallel to y = 3x + 4 is k = 6
