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Are the two given lines parallel , perpendicular, or neither?

a. y=1/4x+1, y=4x-1

b. f(x)= 7x-5, g(x)=x+4

c. y=3, x=4

d. The lines that passes through (-1,2) and (3,-2) and the line -5x+5y=15

Take the two equations as y = Ax + B and y = Cx + D

with the variables in both as x and y.

The letters A, B, C, and D are constants.

The slope of the 1st line is A and the slope of 2nd line is C.

They are parallel if A = C and perpendicular if AC = -1.

a. Here, the slopes of the two lines are A = 1/4 and C = 4.

These are not the parallel since A is not equal to C.

Next, note that AC = 1, not -1, so they are not perpendicular.

Correct answer: neither

b. The slopes of the two lines are A = 7 and C = 1.

They are not parallel since A is not equal to C.

Next, note AC = 7, so they are not perpendicular.

Correct answer: neither

c. The line y=3 is horizontal and A=0, since it is not there.

The line x=4 is vertical, since there is no Y involved.

This is a special case since neither line is an equation of x and y.

If both formulas had y, they would be parallel.

They would also be parallel if both formulas had x.

Since one has x and the other has y, they are perpendicular.

d. The slope of the first line is (y2-y1)/(x2-x1) where (x1,y1) = (-1,2) and (x2,y2) = (3,-2).

That makes the slope 4/4 = 1.

The second line needs to be converted to proper form, with a y alone on one side.

Adding 5x to both sides gives 5y = 5x + 15.

Dividing the entire equation by 5 gives y = x + 3.

Once in this form, the slope is seen to be 1.

Since the slope of the 1st line is 1 and the slope of the 2nd line is 1,

they are parallel.

Algebra

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