Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[x+6y=-6],[2x+13 y=-10]:} No Solutions Infinitely Many Solutions One Solution

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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.  {:[x+6y=-6],[2x+13 y=-10]:} No Solutions Infinitely Many Solutions One Solution

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Analyze Equations: Analyze the system of equations to determine the relationship between the two equations. We have the system: 1) x + 6y = -6 2) 2x + 13y = -10 We will use the method of coefficients to compare the ratios of the coefficients of x, y, and the constants to determine if the lines are parallel, the same, or intersecting.Calculate Ratio of Coefficients (x): Calculate the ratio of the coefficients of x in both equations. For the first equation, the coefficient of x is 1. For the second equation, the coefficient of x is 2. The ratio is 1/2.Calculate Ratio of Coefficients (y): Calculate the ratio of the coefficients of y in both equations. For the first equation, the coefficient of y is 6. For the second equation, the coefficient of y is 13. The ratio is 6/13.Calculate Ratio of Constants: Calculate the ratio of the constants in both equations. For the first equation, the constant is -6. For the second equation, the constant is -10. The ratio is -6/-10, which simplifies to 3/5.Compare Ratios: Compare the ratios calculated in Steps 2, 3, and 4. The ratios of the coefficients of x and y are different (1/2 and 6/13, respectively), which means the lines are not parallel and not the same line. The ratio of the constants (3/5) is also different from the ratios of the coefficients. This indicates that the lines intersect at exactly one point.Conclude Solution Type: Conclude the type of solution the system has based on the comparison. Since the lines intersect at exactly one point, the system of equations has exactly one solution.

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. $\newline$ $\begin{aligned} x+6 y & =-6 \\ 2 x+13 y & =-10 \end{aligned}$ $\newline$ No Solutions $\newline$ Infinitely Many Solutions $\newline$ One Solution

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