Given an equation, y=mx+b, explain how to create an equation for a second line so that the system will have no solutions or infinitely many solutions.

  no solutions: the y-intercept must be changed

  infinite solutions: the equation must describe the same line

Step-by-step explanation:

No Solutions

For the system of equations to have no solutions, the equations must describe parallel lines. The lines will have the same slope (m), but will have different y-intercepts (b).

The second equation could be ...

  y = mx +b1 . . . . . where b1 ≠ b

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Infinitely Many Solutions

The two equations of the system must describe the same line. The second equation must reduce to exactly the same equation as the one given: y = mx +b. It could be this equation multiplied by a constant, for example:

  ky = mkx +kb . . . . . equation multiplied by k

Or, it could be rearranged:

  mx -y +b = 0 . . . . . general form

  mx -y = -b . . . . . . . .standard form

  y -b = mx . . . . . . . . point-slope form; line with slope m through point (0, b)

c. 1,233 miles