Answer:
Mark downloaded 1 pop songs, 4 rock songs, and 5 hip hop songs.
Step-by-step explanation:
Let us assume that the number of pop songs be x .
Let us assume that the number of rock songs be y .
Let us assume the the number of hip hop songs be z .
As given
Mark downloaded 10 songs in total consisting of pop, rock, and hip hop.
Than the equation becomes
x + y + z = 10
Jessica downloaded five times as many pop songs, twice as many rock, and three times as many hip hop songs as Mark.
Number of pop songs jessica download = 5x
Number of rock songs jessica download = 2y
Number of hip hop songs jessica download = 3z
She downloaded 28 songs total.
Than the equation becomes
5x + 2y + 3z = 28
As given
If Nate downloaded 20 songs total .
Three times as many pop songs, three times as many rock songs, and the same number of hip hop songs as Mark .
Number of pop songs Nate download = 3x
Number of rock songs Nate download = 3y
Number of hip hop songs Nate download = z
Than the equation becomes
3x + 3y + z = 20
Thus the three equations are .
x + y + z = 10
5x + 2y + 3z = 28
3x + 3y + z = 20
Multiply x + y + z = 10 by 2 and subtracted from 5x + 2y + 3z = 28 .
5x - 2x + 2y - 2y + 3z - 2z = 28 - 20
3x + z = 8
Multiply x + y + z = 10 by 3 and subtracted from 3x + 3y + z = 20 .
3x - 3x + 3y - 3y + z - 3z = 20 -30
-2z = -10
[tex]z = \frac{-10}{-2}[/tex]
z = 5
Put in the equation 3x + z = 8 .
3x + 5 = 8
3x = 8 - 5
[tex]x = \frac{3}{3}[/tex]
x = 1
Put x = 1 , z = 5 in the equation x + y + z = 10 .
1 + 5 + y = 10
6 + y = 10
y = 10 - 6
y = 4
Thus
Number of pop songs mary download = 1
Number of rock songs mary download = 4
Number of hip hop songs mary download = 5
Therefore Mark downloaded 1 pop songs, 4 rock songs, and 5 hip hop songs.
