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.
