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x=−(3−cy)+6−7xx = -(3 - cy) + 6 - 7x 3x−y=32−x3x - y = \frac{3}{2} - x In the system of equations, cc is a constant. For what value of cc does the system of linear equations have infinitely many solutions?

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Q. x=−(3−cy)+6−7xx = -(3 - cy) + 6 - 7x 3x−y=32−x3x - y = \frac{3}{2} - x In the system of equations, cc is a constant. For what value of cc does the system of linear equations have infinitely many solutions?
  1. Simplify equation: Simplify the first equation: x=−(3−cy)+6−7xx = -(3 - cy) + 6 - 7x.
    Combine like terms and simplify:
    x=−3+cy+6−7xx = -3 + cy + 6 - 7x
    x+7x=cy+3x + 7x = cy + 3
    8x=cy+38x = cy + 3
  2. Combine terms and simplify: Simplify the second equation: 3x−y=32−x3x - y = \frac{3}{2} - x.\newlineMove xx to the left side:\newline3x+x−y=323x + x - y = \frac{3}{2}\newline4x−y=324x - y = \frac{3}{2}
  3. Move xx to left side: To find the value of cc for which the system has infinitely many solutions, the equations must be proportional.\newlineCompare the coefficients from 8x=cy+38x = cy + 3 and 4x−y=324x - y = \frac{3}{2}.\newlineFrom the first equation, the coefficient of yy is cc and from the second equation, it is −1-1.\newlineSet the ratios equal:\newline84=c−1\frac{8}{4} = \frac{c}{-1}\newline2=c−12 = \frac{c}{-1}\newlinec=−2c = -2

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