Answer :
Let's solve the problem step-by-step using the information given:
1. Identify Variables:
- Let [tex]\( x \)[/tex] be the pounds of almonds.
- Let [tex]\( y \)[/tex] be the pounds of cashews.
- Let [tex]\( z \)[/tex] be the pounds of walnuts.
2. Set Up Equations:
- From the problem, we have the following conditions:
a. The total weight of the nuts is 12 pounds:
[tex]\[
x + y + z = 12
\][/tex]
b. The total cost of the nuts is $118:
[tex]\[
7x + 10y + 12z = 118
\][/tex]
c. The customer buys 2 more pounds of walnuts than cashews:
[tex]\[
z = y + 2
\][/tex]
3. Solve the Equations:
- We will use these three equations to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
- Substitute [tex]\( z = y + 2 \)[/tex] from (c) into equations (a) and (b):
[tex]\[
x + y + (y + 2) = 12
\][/tex]
Simplify:
[tex]\[
x + 2y + 2 = 12 \implies x + 2y = 10
\][/tex]
- Now substitute [tex]\( z = y + 2 \)[/tex] into equation (b):
[tex]\[
7x + 10y + 12(y + 2) = 118
\][/tex]
Simplify:
[tex]\[
7x + 10y + 12y + 24 = 118
\][/tex]
[tex]\[
7x + 22y = 94
\][/tex]
4. Solve the Simplified System:
- We have the system of equations:
[tex]\[
x + 2y = 10
\][/tex]
[tex]\[
7x + 22y = 94
\][/tex]
- Solve for [tex]\( x \)[/tex] in the first equation:
[tex]\[
x = 10 - 2y
\][/tex]
- Substitute [tex]\( x = 10 - 2y \)[/tex] into the second equation:
[tex]\[
7(10 - 2y) + 22y = 94
\][/tex]
Simplify:
[tex]\[
70 - 14y + 22y = 94
\][/tex]
[tex]\[
70 + 8y = 94
\][/tex]
[tex]\[
8y = 24
\][/tex]
[tex]\[
y = 3
\][/tex]
- Substitute [tex]\( y = 3 \)[/tex] back into [tex]\( x = 10 - 2y \)[/tex]:
[tex]\[
x = 10 - 2(3) = 4
\][/tex]
- Find [tex]\( z \)[/tex] using [tex]\( z = y + 2 \)[/tex]:
[tex]\[
z = 3 + 2 = 5
\][/tex]
5. Interpret the Results:
- The customer bought 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.
Based on the possible interpretations in the question, none of the given statements exactly match these results. However, with the correct results [tex]\( x = 4 \)[/tex], [tex]\( y = 3 \)[/tex], and [tex]\( z = 5 \)[/tex], we can understand the solution of the nut quantities purchased by the customer.
1. Identify Variables:
- Let [tex]\( x \)[/tex] be the pounds of almonds.
- Let [tex]\( y \)[/tex] be the pounds of cashews.
- Let [tex]\( z \)[/tex] be the pounds of walnuts.
2. Set Up Equations:
- From the problem, we have the following conditions:
a. The total weight of the nuts is 12 pounds:
[tex]\[
x + y + z = 12
\][/tex]
b. The total cost of the nuts is $118:
[tex]\[
7x + 10y + 12z = 118
\][/tex]
c. The customer buys 2 more pounds of walnuts than cashews:
[tex]\[
z = y + 2
\][/tex]
3. Solve the Equations:
- We will use these three equations to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
- Substitute [tex]\( z = y + 2 \)[/tex] from (c) into equations (a) and (b):
[tex]\[
x + y + (y + 2) = 12
\][/tex]
Simplify:
[tex]\[
x + 2y + 2 = 12 \implies x + 2y = 10
\][/tex]
- Now substitute [tex]\( z = y + 2 \)[/tex] into equation (b):
[tex]\[
7x + 10y + 12(y + 2) = 118
\][/tex]
Simplify:
[tex]\[
7x + 10y + 12y + 24 = 118
\][/tex]
[tex]\[
7x + 22y = 94
\][/tex]
4. Solve the Simplified System:
- We have the system of equations:
[tex]\[
x + 2y = 10
\][/tex]
[tex]\[
7x + 22y = 94
\][/tex]
- Solve for [tex]\( x \)[/tex] in the first equation:
[tex]\[
x = 10 - 2y
\][/tex]
- Substitute [tex]\( x = 10 - 2y \)[/tex] into the second equation:
[tex]\[
7(10 - 2y) + 22y = 94
\][/tex]
Simplify:
[tex]\[
70 - 14y + 22y = 94
\][/tex]
[tex]\[
70 + 8y = 94
\][/tex]
[tex]\[
8y = 24
\][/tex]
[tex]\[
y = 3
\][/tex]
- Substitute [tex]\( y = 3 \)[/tex] back into [tex]\( x = 10 - 2y \)[/tex]:
[tex]\[
x = 10 - 2(3) = 4
\][/tex]
- Find [tex]\( z \)[/tex] using [tex]\( z = y + 2 \)[/tex]:
[tex]\[
z = 3 + 2 = 5
\][/tex]
5. Interpret the Results:
- The customer bought 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.
Based on the possible interpretations in the question, none of the given statements exactly match these results. However, with the correct results [tex]\( x = 4 \)[/tex], [tex]\( y = 3 \)[/tex], and [tex]\( z = 5 \)[/tex], we can understand the solution of the nut quantities purchased by the customer.
