Question
2x+3y=-6
4x+6y=-12 have a solution
4x+6y=-12 have a solution
Answers
GPT-4o mini
To determine if the two equations have a solution, we first assess whether they are consistent or inconsistent and if they represent the same line or different lines.
The given equations are:
1. \(2x + 3y = -6\) (Equation 1)
2. \(4x + 6y = -12\) (Equation 2)
Let's simplify Equation 2 to see if it is a multiple of Equation 1:
Dividing Equation 2 by 2:
\[
\frac{4x + 6y}{2} = \frac{-12}{2}
\]
This simplifies to:
\[
2x + 3y = -6
\]
Notice that after dividing Equation 2 by 2, we get the same equation as Equation 1. This means that both equations represent the same line.
Since they represent the same line, there are infinitely many solutions to this system of equations, represented as points along the line defined by either equation.
In conclusion, **the system of equations has infinitely many solutions**.
The given equations are:
1. \(2x + 3y = -6\) (Equation 1)
2. \(4x + 6y = -12\) (Equation 2)
Let's simplify Equation 2 to see if it is a multiple of Equation 1:
Dividing Equation 2 by 2:
\[
\frac{4x + 6y}{2} = \frac{-12}{2}
\]
This simplifies to:
\[
2x + 3y = -6
\]
Notice that after dividing Equation 2 by 2, we get the same equation as Equation 1. This means that both equations represent the same line.
Since they represent the same line, there are infinitely many solutions to this system of equations, represented as points along the line defined by either equation.
In conclusion, **the system of equations has infinitely many solutions**.