Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[x+6y=8],[-x-6y=-8]:} No Solutions One Solution Infinitely Many Solutions

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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.  {:[x+6y=8],[-x-6y=-8]:} No Solutions One Solution Infinitely Many Solutions

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Analyze System of Equations: When analyzing a system of linear equations to determine the number of solutions, we need to look at the coefficients of the variables and the constants. If the two equations are multiples of each other, then they represent the same line and have infinitely many solutions. If they have the same coefficients but different constants, they are parallel and have no solutions. If they have different coefficients, they intersect at one point and have exactly one solution.Compare Coefficients and Constants: Let's compare the coefficients of the variables and the constants in the two equations. The first equation is x + 6y = 8, and the second equation is -x - 6y = -8. We can see that the coefficients of x and y in the second equation are the negatives of those in the first equation, and the constant term is also the negative of the constant in the first equation.Identify Same Line Representation: If we multiply the entire first equation by -1, we would get -x - 6y = -8, which is exactly the same as the second equation. This means that the two equations are actually the same line represented in different ways. Therefore, the system of equations has infinitely many solutions because every point on the line is a solution to both equations.

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. $\newline$ $\begin{aligned} x+6 y & =8 \\ -x-6 y & =-8 \end{aligned}$ $\newline$ No Solutions $\newline$ One Solution $\newline$ Infinitely Many Solutions

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