Final answer:
The problem involves creating two linear equations based on the given information and then solving them simultaneously to find that there are six 6-volt batteries and four 12-volt batteries among the ten.
Step-by-step explanation:
Solving the Battery Voltage Problem
To solve the problem, we will use a system of linear equations. Let's denote the number of 6-volt batteries as x and the number of 12-volt batteries as y. Since there are ten batteries in total, we can write the following equation:
1. x + y = 10 (Equation for the total number of batteries)
The sum of voltages of all batteries is given to be 84 volts. Each 6-volt battery contributes 6 volts and each 12-volt battery contributes 12 volts to the total voltage. We can write the second equation for the sum of the voltages:
2. 6x + 12y = 84 (Equation for the total voltage from the batteries)
To find the values of x and y, we can simplify the second equation:
3. Divide the second equation by 6:
x + 2y = 14
Now we can subtract the first equation from this new equation to find the value of y:
4. (x + 2y) - (x + y) = 14 - 10
5. y = 4
Now that we know there are four 12-volt batteries, we can use the first equation to find the number of 6-volt batteries:
6. x = 10 - y
7. x = 10 - 4
8. x = 6
Therefore, the group consists of six 6-volt batteries and four 12-volt batteries.