Question
Which describes the system of equations below?
7x + 8y = 5
14x + 16y = 10
7x + 8y = 5
14x + 16y = 10
Answers
Answered by
GPT-4o mini
To analyze the given system of equations:
1. **First equation**: \(7x + 8y = 5\)
2. **Second equation**: \(14x + 16y = 10\)
Let's see if the second equation is a multiple of the first equation.
Notice that if we multiply the first equation by 2, we get:
\[
2(7x + 8y) = 2(5)
\]
which simplifies to:
\[
14x + 16y = 10
\]
This is exactly the same as the second equation. Since the second equation can be derived from the first by multiplying by a constant, these two equations represent the same line.
### Conclusion:
The system of equations has infinitely many solutions because both equations represent the same line in a two-dimensional plane. Any point that satisfies one equation will also satisfy the other.
In summary, this system of equations is **dependent** and has **infinitely many solutions**.
1. **First equation**: \(7x + 8y = 5\)
2. **Second equation**: \(14x + 16y = 10\)
Let's see if the second equation is a multiple of the first equation.
Notice that if we multiply the first equation by 2, we get:
\[
2(7x + 8y) = 2(5)
\]
which simplifies to:
\[
14x + 16y = 10
\]
This is exactly the same as the second equation. Since the second equation can be derived from the first by multiplying by a constant, these two equations represent the same line.
### Conclusion:
The system of equations has infinitely many solutions because both equations represent the same line in a two-dimensional plane. Any point that satisfies one equation will also satisfy the other.
In summary, this system of equations is **dependent** and has **infinitely many solutions**.