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How many solutions does the system have? 12x - 15y = 18
4x - 5y = 6
4x - 5y = 6
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Answered by
GPT-4o mini
To determine how many solutions the system of equations has, we can analyze the two equations given:
1. \( 12x - 15y = 18 \)
2. \( 4x - 5y = 6 \)
A common method is to simplify the equations and compare their forms.
First, observe the first equation:
\[
12x - 15y = 18
\]
We can simplify this by dividing all terms by 3:
\[
4x - 5y = 6
\]
Now we see that the first equation, after simplification, actually becomes identical to the second equation:
\[
4x - 5y = 6
\]
Since both equations are the same, they represent the same line in a two-dimensional space. Therefore, there are infinitely many solutions, as any point on this line is a solution to the system.
In conclusion, the system has **infinitely many solutions**.
1. \( 12x - 15y = 18 \)
2. \( 4x - 5y = 6 \)
A common method is to simplify the equations and compare their forms.
First, observe the first equation:
\[
12x - 15y = 18
\]
We can simplify this by dividing all terms by 3:
\[
4x - 5y = 6
\]
Now we see that the first equation, after simplification, actually becomes identical to the second equation:
\[
4x - 5y = 6
\]
Since both equations are the same, they represent the same line in a two-dimensional space. Therefore, there are infinitely many solutions, as any point on this line is a solution to the system.
In conclusion, the system has **infinitely many solutions**.
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