Always location and name your variables first. So, let's call:
- The number of books Ky has: #k#
- The number of books Grant has: #g#
- The number of books Jamie has: #j#
Next, we can write the three equations from the information in the problem:
Equation 1: #k = 3g#
Equation 2: #g = j - 6#
Equation 3: #k + g + j = 176#
First, solve Equation 2 for #j#:
#g = j - 6#
#g + color(red)(6) = j - 6 + color(red)(6)#
#g + 6 = j - 0#
#g + 6 = j#
#j = g + 6#
Next, using this result we can substitute #(g + 6)# for #j# in Equation 3. And using Equation 1 we can substitute #3g# for #k# into Equation 3. Then we can solve Equation 3 for #g#:
#k + g + j = 176# becomes:
#3g + g + (g + 6) = 176#
#3g + g + g + 6 = 176#
#3g + g + g + 6 - color(red)(6) = 176 - color(red)(6)#
#3g + g + g + 0 = 170#
#3g + g + g = 170#
#3g + 1g + 1g = 170#
#(3 + 1 + 1)g = 170#
#5g = 170#
#(5g)/color(red)(5) = 170/color(red)(5)#
#(color(red)(cancel(color(black)(5)))g)/cancel(color(red)(5)) = 34#
#g = 34#
Therefore, Grant has #color(red)(34)# books.
Now, substitute #34# for #g# in the solution for #j# we did previously and calculate the number of books Jamie has:
#j = g + 6# becomes:
#j = 34 + 6 = 40#
Jamie has #color(red)(40)# books
We can also calculate the number of books Ky has by substituting #34# for #g# in Equation 1 and calculating #k#:
#k = 3g# becomes:
#k = 3 xx 34 = 102#
Ky has #color(red)(102)# books
#k + g + j = 102 + 34 + 40 = 176#
