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Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

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BuyFindarrow_forward

Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Without graphing, determine whether the pair of lines are parallel, perpendicular, the same, or none of these:

a) x + 3 y = 6 and 3 x - y = - 7

b) 2 x - y = - 3 and y = 2 x - 14

c) y + 2 = - 3 x - 5 and 2 y = 6 x + 11

d) 0.5 x + y = 0 and 2 x - y = 10

To determine

(a)

To check:

The relationship between the lines x+3y=6 and 3x-y=-7.

Explanation

Initially check the given equation,

If the one equation is derived from other, then these lines are same or coincident in nature.

The two given equations x+3y=6 and 3x-y=-7 are not derived from one another.

If not, then proceed with the following procedures.

By theorem,

If two lines are parallel, then their slopes are equal.

(i.e) If l1l2, then m1=m2.

If two lines are perpendicular, then the product of their slopes is -1 or one slope is negative reciprocal of the other slope.

(i.e) If l1l2, then m1.m2=-1 or m2=-1m1

To find the relation between the lines, the slopes of the lines are needed.

Thus we need to find the points on the lines.

Let m1 and m2 be the slopes of the lines x+3y=6 and 3x-y=-7 respectively.

The slope of the line that contains the points x1,y1 and x2,y2 is given by

m=y2-y1x2-x1 for x2x1

The points of the line x+3y=6 is determined by x-intercept and y-intercept.

Substitute x=0 in the given equation,

0+3y=6

On simplifying,

3y=6

The coefficient of y is 3, thus divide both sides by 3.

3y3=63

y=2

Thus the first point is 0, 2.

Substitute y=0 in the given equation,

x+30=6

x=6

Thus the second point is 6, 0.

The line x+3y=6 containing the points are 0, 2 and 6, 0.

Using slope formula and choosing x1=0, y1=2, x2=6, and y2=0,

Slope m1=0-26-0

m1=-26

m1=-13

The points of the line 3x-y=-7 is determined by x-intercept and y-intercept

To determine

(b)

To check:

The relationship between the lines 2x-y=-3 and y=2x-14.

To determine

(c)

To check:

The relationship between the lines y+2=-3x-5 and 2y=6x+11.

To determine

(d)

To check:

The relationship between the lines 0.5x+y=0 and 2x-y=10.

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