Systems of Linear Equations and Their Solutions

Question

Mai is given these two systems of linear equations to solve:

She analyzed them for a moment, and then—without graphing the equations—said, "I got it! One of the systems has no solution and the other has infinitely many solutions!" Mai is right!

Which system has no solution and which one has many solutions? Explain or show how you know (without graphing the equations). 

Answer 1

System #1:

-4 * equation 1 + equation 2...

0 = 12

NO Solution

System #2:

solves equation #1 for y,

y = 13 - 5x

20x = 52 - 4(13 - 5x)

20x = 52 - 52 + 20x

20x = 20x

0 = 0

All ordered pairs

check:

y = 3 ---> x=2

20*2 = 52 - 4*3

40 = 52 - 12

yes

y=8 --> x=1

20*1 = 52 - 4*8

20 = 52 - 32

yes

Answer 2

Multiply the first original equation by -4 on both systems.

-20x - 4y = -52 -20x - 4y = -52

20x + 4y = 64 20x + 4y = 52

0 ≠ 12 0 = 0

System 2 has infinitely many solutions because both lines are the same. System 1 has no solutions because both lines are parallel.

Answer 3

system 1 has no solutions. If you multiply the top equation by 4 you get 20x +4y = 52. The bottom equation says 20x +4y =64

20x +4y cannot equal both 52 and 64