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Math
GeometryElementary Geometry for College Students6th Edition

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Chapter 10.2, Problem 23E
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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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In Exercises 21 to 24, state whether the lines are parallel, perpendicular, the same (coincident), or none of these.

2 x + 3 y = 6 and 3 x - 2 y = 12

To determine

To check:

The relationship between the lines 2x+3y=6 and 3x-2y=12.

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 2x+3y=6 and 3x-2y=12 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 l1∥l2, 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 l1⊥l2, 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 2x+3y=6 and 3x-2y=12 respectively.

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

m=y2-y1x2-x1 for x2≠x1

The points of the line 2x+3y=6 is determined by x-intercept and y-intercept.

Substitute x=0 in the given equation,

20+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,

2x+30=6

2x=6

The coefficient of x is 2, thus divide both sides by 2.

2x2=62

On simplifying,

x=3

Thus the second point is 3, 0.

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

Using slope formula and choosing x1=0, y1=2, x2=3, and y2=0,

Slope m1=0-23-0

m1=-23

The points of the line 3x-2y=12 is determined by x-intercept and y-intercept

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