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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[-x+2y=-5],[-4x+8y=-23]:} Infinitely Many Solutions One Solution No Solutions

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline−x+2yamp;=−5−4x+8yamp;=−23 \begin{aligned} -x+2 y & =-5 \\ -4 x+8 y & =-23 \end{aligned} \newlineInfinitely Many Solutions\newlineOne Solution\newlineNo Solutions

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Q. Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline−x+2y=−5−4x+8y=−23 \begin{aligned} -x+2 y & =-5 \\ -4 x+8 y & =-23 \end{aligned} \newlineInfinitely Many Solutions\newlineOne Solution\newlineNo Solutions
  1. Check Coefficients Multiples: We are given the system of equations:\newline11) −x+2y=−5-x + 2y = -5\newline22) −4x+8y=−23-4x + 8y = -23\newlineThe first step is to check if the two equations are multiples of each other, which would indicate that they are essentially the same line and would have infinitely many solutions if the constants also have the same ratio, or no solutions if they do not.
  2. Find Coefficients Ratio: Let's find the ratio of the coefficients of xx and yy in both equations.\newlineFor the first equation, the coefficients are −1-1 for xx and 22 for yy.\newlineFor the second equation, the coefficients are −4-4 for xx and 88 for yy.\newlineThe ratio of the coefficients of xx is yy11.\newlineThe ratio of the coefficients of yy is yy33.\newlineSince both ratios are equal, the lines are parallel or the same line. We need to check the constants to determine which case it is.
  3. Compare Constants Ratios: Now let's compare the ratio of the constants in both equations.\newlineThe constant in the first equation is −5-5 and in the second equation is −23-23.\newlineThe ratio of the constants is −23/−5-23 / -5, which is not equal to 44 (the ratio of the coefficients).\newlineSince the ratios of the coefficients are the same, but the ratio of the constants is different, the lines are parallel and do not intersect. This means the system of equations has no solutions.

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