Answer :
Let's denote the weight of one bag of almonds as [tex]\( A \)[/tex] grams and the weight of one bag of peanuts as [tex]\( P \)[/tex] grams.
We are provided with two key pieces of information that lead to the following system of equations:
1. [tex]\( 2A + 7P = 760 \)[/tex]
2. [tex]\( 4A + 5P = 980 \)[/tex]
We need to solve this system of equations to find the values of [tex]\( A \)[/tex] and [tex]\( P \)[/tex].
### Solving Step-by-Step
1. Equation Setup:
- First equation: [tex]\( 2A + 7P = 760 \)[/tex]
- Second equation: [tex]\( 4A + 5P = 980 \)[/tex]
2. Express one variable in terms of the other:
Let's solve the first equation for [tex]\( A \)[/tex]:
[tex]\[ 2A + 7P = 760 \implies A = \frac{760 - 7P}{2} \][/tex]
3. Substitute into the second equation:
Substitute [tex]\( A \)[/tex] from the first equation into the second equation:
[tex]\[ 4\left(\frac{760 - 7P}{2}\right) + 5P = 980 \][/tex]
4. Simplify the equation:
First, simplify the left-hand side:
[tex]\[ 4 \times \frac{760 - 7P}{2} = 2(760 - 7P) = 1520 - 14P \][/tex]
Thus the equation becomes:
[tex]\[ 1520 - 14P + 5P = 980 \][/tex]
Simplifying further:
[tex]\[ 1520 - 9P = 980 \][/tex]
5. Solve for [tex]\( P \)[/tex]:
[tex]\[ 1520 - 980 = 9P \][/tex]
[tex]\[ 540 = 9P \][/tex]
[tex]\[ P = \frac{540}{9} = 60 \][/tex]
So, the weight of one bag of peanuts, [tex]\( P \)[/tex], is 60 grams.
6. Substitute [tex]\( P \)[/tex] back into the first equation to find [tex]\( A \)[/tex]:
[tex]\[ 2A + 7(60) = 760 \][/tex]
[tex]\[ 2A + 420 = 760 \][/tex]
[tex]\[ 2A = 760 - 420 \][/tex]
[tex]\[ 2A = 340 \][/tex]
[tex]\[ A = \frac{340}{2} = 170 \][/tex]
So, the weight of one bag of almonds, [tex]\( A \)[/tex], is 170 grams.
### Conclusion
Each bag of almonds weighs 170 grams, and each bag of peanuts weighs 60 grams.
We are provided with two key pieces of information that lead to the following system of equations:
1. [tex]\( 2A + 7P = 760 \)[/tex]
2. [tex]\( 4A + 5P = 980 \)[/tex]
We need to solve this system of equations to find the values of [tex]\( A \)[/tex] and [tex]\( P \)[/tex].
### Solving Step-by-Step
1. Equation Setup:
- First equation: [tex]\( 2A + 7P = 760 \)[/tex]
- Second equation: [tex]\( 4A + 5P = 980 \)[/tex]
2. Express one variable in terms of the other:
Let's solve the first equation for [tex]\( A \)[/tex]:
[tex]\[ 2A + 7P = 760 \implies A = \frac{760 - 7P}{2} \][/tex]
3. Substitute into the second equation:
Substitute [tex]\( A \)[/tex] from the first equation into the second equation:
[tex]\[ 4\left(\frac{760 - 7P}{2}\right) + 5P = 980 \][/tex]
4. Simplify the equation:
First, simplify the left-hand side:
[tex]\[ 4 \times \frac{760 - 7P}{2} = 2(760 - 7P) = 1520 - 14P \][/tex]
Thus the equation becomes:
[tex]\[ 1520 - 14P + 5P = 980 \][/tex]
Simplifying further:
[tex]\[ 1520 - 9P = 980 \][/tex]
5. Solve for [tex]\( P \)[/tex]:
[tex]\[ 1520 - 980 = 9P \][/tex]
[tex]\[ 540 = 9P \][/tex]
[tex]\[ P = \frac{540}{9} = 60 \][/tex]
So, the weight of one bag of peanuts, [tex]\( P \)[/tex], is 60 grams.
6. Substitute [tex]\( P \)[/tex] back into the first equation to find [tex]\( A \)[/tex]:
[tex]\[ 2A + 7(60) = 760 \][/tex]
[tex]\[ 2A + 420 = 760 \][/tex]
[tex]\[ 2A = 760 - 420 \][/tex]
[tex]\[ 2A = 340 \][/tex]
[tex]\[ A = \frac{340}{2} = 170 \][/tex]
So, the weight of one bag of almonds, [tex]\( A \)[/tex], is 170 grams.
### Conclusion
Each bag of almonds weighs 170 grams, and each bag of peanuts weighs 60 grams.
