Asked by p
How many solutions does the system have? 12x−15y=18 4x−5y=6 (1 point) Responses infinitely many solutions infinitely many solutions two solutions two solutions no solution no solution one solution
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Answered by
GPT-4o mini
To determine how many solutions the system of equations has, we can analyze the given equations:
1. \( 12x - 15y = 18 \)
2. \( 4x - 5y = 6 \)
First, let's simplify both equations.
Starting with the second equation:
\[
4x - 5y = 6
\]
We can multiply the entire equation by 3 to help us compare it to the first equation:
\[
3(4x - 5y) = 3 \cdot 6
\]
\[
12x - 15y = 18
\]
Now we can see that the second equation \( 4x - 5y = 6 \) is simply a scaled version of the first equation \( 12x - 15y = 18 \).
Since both equations represent the same line, they have infinitely many solutions; every point on the line \( 12x - 15y = 18 \) (or equivalently \( 4x - 5y = 6 \)) is a solution to the system.
Thus, the answer is **infinitely many solutions**.
1. \( 12x - 15y = 18 \)
2. \( 4x - 5y = 6 \)
First, let's simplify both equations.
Starting with the second equation:
\[
4x - 5y = 6
\]
We can multiply the entire equation by 3 to help us compare it to the first equation:
\[
3(4x - 5y) = 3 \cdot 6
\]
\[
12x - 15y = 18
\]
Now we can see that the second equation \( 4x - 5y = 6 \) is simply a scaled version of the first equation \( 12x - 15y = 18 \).
Since both equations represent the same line, they have infinitely many solutions; every point on the line \( 12x - 15y = 18 \) (or equivalently \( 4x - 5y = 6 \)) is a solution to the system.
Thus, the answer is **infinitely many solutions**.
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