   # Further Investigations There is another way of telling whether a system of two linear equations in two unknowns is consistent or inconsistent, or whether the equations are dependent, without taking the time to graph each equation. It can be shown that any system of the form a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 Has one and only one solution if a 1 a 2 ≠ b 1 b 2 That it has no solution if a 1 a 2 = b 1 b 2 ≠ c 1 c 2 And that it has infinitely many solutions if a 1 a 2 = b 1 b 2 = c 1 c 2 For each of the following systems, determine whether the system is consistent, the system is inconsistent, or the equations are dependent. (a) ( 4 x − 3 y = 7 9 x + 2 y = 5 ) (b) ( 5 x − y = 6 10 x − 2 y = 19 ) (c) ( 5 x − 4 y = 11 4 x + 5 y = 12 ) (d) ( x + 2 y = 5 x − 2 y = 9 ) (e) ( x − 3 y = 5 3 x − 9 y = 15 ) (f) ( 4 x + 3 y = 7 2 x − y = 10 ) (g) ( 3 x + 2 y = 4 y = − 3 2 x − 1 ) (h) ( y = 4 3 x − 2 4 x − 3 y = 6 ) ### Intermediate Algebra

10th Edition
Jerome E. Kaufmann + 1 other
Publisher: Cengage Learning
ISBN: 9781285195728

#### Solutions

Chapter
Section ### Intermediate Algebra

10th Edition
Jerome E. Kaufmann + 1 other
Publisher: Cengage Learning
ISBN: 9781285195728
Chapter 10.3, Problem 64PS
Textbook Problem
1 views

## Further InvestigationsThere is another way of telling whether a system of two linear equations in two unknowns is consistent or inconsistent, or whether the equations are dependent, without taking the time to graph each equation. It can be shown that any system of the form a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 Has one and only one solution if a 1 a 2 ≠ b 1 b 2 That it has no solution if a 1 a 2 = b 1 b 2 ≠ c 1 c 2 And that it has infinitely many solutions if a 1 a 2 = b 1 b 2 = c 1 c 2 For each of the following systems, determine whether the system is consistent, the system is inconsistent, or the equations are dependent. (a) ( 4 x − 3 y = 7 9 x + 2 y = 5 ) (b) ( 5 x − y = 6 10 x − 2 y = 19 ) (c) ( 5 x − 4 y = 11 4 x + 5 y = 12 ) (d) ( x + 2 y = 5 x − 2 y = 9 ) (e) ( x − 3 y = 5 3 x − 9 y = 15 ) (f) ( 4 x + 3 y = 7 2 x − y = 10 ) (g) ( 3 x + 2 y = 4 y = − 3 2 x − 1 ) (h) ( y = 4 3 x − 2 4 x − 3 y = 6 )

To determine

(a)

Whether the system is consistent, the system is inconsistent, or the equations are dependent.

### Explanation of Solution

Approach:

The system of the form,

a1x+b1y=c1a2x+b2y=c2

i) Has one and only one solution if,

a1a2b1b2 (Consistent).

ii) That it has no solution if,

a1a2=b1b2c1c2 (Inconsistent).

iii) And that it has infinitely many solutions if,

a1a2=b1b2=c1c2 (Equations are dependent)

To determine

(b)

Whether the system is consistent, the system is inconsistent, or the equations are dependent.

To determine

(c)

Whether the system is consistent, the system is inconsistent, or the equations are dependent.

To determine

(d)

Whether the system is consistent, the system is inconsistent, or the equations are dependent.

To determine

(e)

Whether the system is consistent, the system is inconsistent, or the equations are dependent.

To determine

(f)

Whether the system is consistent, the system is inconsistent, or the equations are dependent.

To determine

(g)

Whether the system is consistent, the system is inconsistent, or the equations are dependent.

To determine

(h)

Whether the system is consistent, the system is inconsistent, or the equations are dependent.

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