Question

A DJ has a total of 1075 dance, rock, and country songs on her system. The dance selection is three times the size of the rock selection. The country selection has 105 more songs than the rock selection. How many songs on the system are dance? rock? country? Write it as a substitution equation.

Answer 1

The DJ has 582 dance, 194 rock and 299 country music on her system

Data;

• Total number of songs = 1075
• dance = x
• rock = y
• country = z

System of Equations

To solve this word problem, let's write systems of equations for this and solve simultaneously by substitution method.

$x+y+z=1075...equation(i)\\x=3y...equation(ii)\\z=105+y...equation(iii)$

from equation (ii)

$x=3y\\y=x/3...equation(iv)$

put equation (iv) into equation (iii)

$z=105+y\\z=105 + \frac{x}{3}...equation(v)$

substitute equation (v) and equation (iv) into equation (i)

$x+y+z=1075\\x+\frac{x}{3}+ (105 + \frac{x}{3})=1075\\ \frac{5}{3}x+105=1075$

solve for x

$5/3x = 1075-105\\5/3x=970\\5x = 970*3\\5x=2910\\5x/5=2910/5\\\\x=582$

Put x = 582 into equation (ii)

$x=3y\\582=3y\\y=582/3\\y=194$

Put y = 194 into equation 3

$z=105+y\\z=105+194\\z=299$

From the calculation above, the DJ has 582 dance, 194 rock and 299 country music on her system

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