### Instruction: $5$ green balls and $2$ red balls together weigh $10$ pounds, and $1$ green ball and $4$ red balls together weigh $7$ pounds. If all red balls weigh the same amount and all green balls weigh the same, then what is the weight of $8$ red and $8$ green balls together?### Response: This problem involves a system of linear equations with two variables: the weight of one green ball and the weight of one red ball.
Let me use $g$ to represent the weight of one green ball and $r$ to represent the weight of one red ball.
Then I can translate the given information into two equations: $5g + 2r = 10$ and $g + 4r = 7$.
I want to solve this system of equations for $g$ and $r$ so that I can find the weight of $8$ green and $8$ red balls together.
One way to do that is to use the elimination method, where I multiply one equation by a constant and then add or subtract it from the other equation to eliminate one variable.
For example, if I multiply the second equation by $-5$, I get $-5g - 20r = -35$.
Then if I add this to the first equation, I get $-18r = -25$.
Dividing both sides by $-18$, I get $r = 25/18$.
This means that one red ball weighs $25/18$ pounds.
Now I can plug this value of $r$ into either equation to find $g$.
For example, using the first equation, I get $5g + 2(25/18) = 10$.
Simplifying, I get $5g = 10 - 50/18 = 180/18 - 50/18 = 130/18$.
Dividing both sides by $5$, I get $g = 130/90 = 13/9$.
This means that one green ball weighs $13/9$ pounds.
Now I have both $g$ and $r$, so I can find the weight of $8$ green and $8$ red balls together.
I just need to multiply $g$ by $8$ and $r$ by $8$ and add them up.
That gives me $8g + 8r = 8(13/9) + 8(25/18)$.
Simplifying, I get $8g + 8r = 104/9 + 100/9 = 204/9$.
This means that $8$ green and $8$ red balls together weigh $204/9$ pounds.