Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[-x+2y=-1],[5x-10 y=5]:} One Solution Infinitely Many Solutions No Solutions

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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.  {:[-x+2y=-1],[5x-10 y=5]:} One Solution Infinitely Many Solutions No Solutions

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Write Equations: Write down the system of equations. We have the system: {-x + 2y = -1, 5x - 10y = 5}Simplify Equations: Look for a way to simplify or manipulate the equations to make them easier to compare. Notice that the second equation can be simplified by dividing all terms by 5. Simplified second equation: x - 2y = 1Compare Equations: Compare the simplified second equation with the first equation. The first equation is: -x + 2y = -1 The simplified second equation is: x - 2y = 1 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 constants are also negatives of each other.Determine Relationship: Determine the relationship between the two equations. Since the two equations are multiples of each other (by -1), they represent the same line. Therefore, every solution to one equation is also a solution to the other.Conclude Solutions: Conclude the number of solutions the system has. Because the two equations represent the same line, there are infinitely many points that satisfy both equations. Hence, the system has infinitely many solutions.

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. $\newline$ $\begin{aligned} -x+2 y & =-1 \\ 5 x-10 y & =5 \end{aligned}$ $\newline$ One Solution $\newline$ Infinitely Many Solutions $\newline$ No Solutions

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