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y = 3x + 4

y = 3x + 4. 2x + y = -1. 2x + y = -1. (-1, 1) is where the two lines intersect. This point is a point on both lines. Therefore, if we substitute -1 in for x and 1 in for y, we should get a true statement, for both equations. y = 3x + 4 1 = 3 (-1) + 4 1 = -3 + 4 = 1

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y = 3x + 4

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  1. y = 3x + 4

  2. 2x + y = -1

  3. 2x + y = -1

  4. (-1, 1) is where the two lines intersect. • This point is a point on both lines. • Therefore, if we substitute -1 in for x and 1 in for y, we should get a true statement, for both equations.

  5. y = 3x + 4 1 = 3 (-1) + 4 1 = -3 + 4 = 1 2x + y = -1 2(-1)+ 1 = -1 -2 + 1 = -1 -1 = -1

  6. The solution to this system of equations is the POINT where the two lines intersect.

  7. What else could happen?

  8. 2x + y = -1

  9. 2x + y = 7

  10. 2x + y = -1 2x + y = 7

  11. We cannot see an intersection for these two line. • The lines are parallel. • These two lines have no points in common. • Therefore, there are no values for x and y, that will make both equations true…2x +y cannot equal -1 and 7 simultaneously.

  12. There is NO SOLUTION to this system of equations.

  13. What else could happen?

  14. y = 3x + 4

  15. 3x – y = -4

  16. 3x – y = -4

  17. Where do these lines intersect? • They intersect at EVERY POINT!! • These two lines have ALL points in common. • Therefore, every point on either line, is also a point on the other line.

  18. There are INFINITELY MANY SOLUTIONS to this system of equations.

  19. 3 Possible Solutions to a System of Equations • Ordered Pair • No Solution • Infinitely Many Solutions The lines intersect at a POINT. The lines are PARALLEL. The equations represent the SAME LINE.

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