582 dance songs, 194 rock songs, and 299 country songs
Practice makes perfect
We want to know how many of the songs the DJ has in her system are dance, rock, and country. Let's create a system of linear equations and then solve it using the Substitution Method. We will do these things one at a time, starting with forming a system. We will use x for the number of dance songs, y for the rock songs, and z for the country songs.
Dance songs:& x
Rock songs:& y
Country songs:& z
From the exercise, we know that there are 1075 songs in total in the system. That means that the sum of all three different song types is 1075. Let's write this fact as an equation.
x+ y+ z= 1075We also know that the dance selection is three times the rock selection and that there are 105 more country songs than there are rock songs. Let's represent these relations using two equations.
x=3 y z= y+ 105
We can combine these three equations into one large system. Let's do it!
x+y+z=1075 & (I) x=3y & (II) z=y+105 & (III)
The system might look complex at first glance, but we can solve it just like a system with two equations. We first look to see if any of the equations are already solved for one of the variables. Equation (II) is solved for x and Equation (III) is solved for z. If we substitute 3y for x and y+105 for z in the first equation, we can eliminate all but the y variable. Let's do it!
Now that we have a value for y, we can substitute its value into any of the equations and solve for another variable. Let's substitute 194 for y in Equation (II) and solve for x.
Now we have a value for x. Good job! Finally, we will repeat the same thing we did to find x, but this time to find z. Let's substitute y= 194 into Equation (III) to find z.
