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Systems of Linear Equations

Systems of Linear Equations. Determinants Graphs of Parallel Lines Graphs of the Same Line Graphs of Intersection Practice. Determinants. What is the use of the determinant? The determinant tells us if there is no solutions/infinitely many or exactly one solution.

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Systems of Linear Equations

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  1. Systems of Linear Equations Determinants Graphs of Parallel LinesGraphs of the Same Line Graphs of Intersection Practice

  2. Determinants • What is the use of the determinant? • The determinant tells us if there is no solutions/infinitely many or exactly one solution. • Given this system of equations, what is the determinant? • ax+by=ecx+dy=f • The determinant is a*d – b*c

  3. Determinants • What is the determinants of the following systems of linear equations? • x= -9-3x=4 • -8x+3y=-245x+2y=15 • 4x-5y=-136x-7y=-5

  4. x= -9-3x=4 (-8)*2 – 3*5 = -31 • -8x+3y=-245x+2y=15 4*(-7) – (-5)*6 = 58 • 4x-5y=-136x-7y=-5 1*0 – (-3)*0 = 0

  5. Determinants • Now what does the determinant number mean? • IF the determinant (a*d – b*c) is ZERO • THEN the system of linear equations has no solution or it has infinitely many. • This means that the lines are either PARALLEL or they are the SAME line. • IF the determinant (a*d – b*c) is NON-ZERO • THEN the system of linear equations has exactly one solution. • This means that the two lines intersect!

  6. Parallel Lines • Parallel lines will have a determinant of ZERO. • Take this system of linear equations • -6x+3y=9-12x+6y=-6 • (-6)*6 – 3*(-12)= 0 • This system is graphedto the right.

  7. Same Line • Systems of linear equations that are the same line have a determinant of ZERO • Take the system, for example. • 2x+3y=94x+6y=18 • 2*6 – 3*4 = 0 • This system is graphedto the right

  8. Intersection • Systems of linear equations that intersect have a determinant that is NON-ZERO. • Take this system, for example. • 4x+6y=-36x-2y=2 • 4*(-2) – 6*6=-44 • This system is graphedto the right.

  9. Practice: How many solutions? 4x-5y=196x+6y=8 3x+6y=10 -2x+4y=3 8x-3y=2 2x+7y=9 15x-5y=25 5x+20y=47 4*6 – (-5)*6=54 3*4 – 6*(-2)=0 8*7 – (-3)*2=62 15*20-(-5)*5=325

  10. Practice: How many solutions? 4x-5y=196x+6y=8 3x+6y=10 -2x+4y=3 8x-3y=2 2x+7y=9 15x-5y=25 5x+20y=47 No Solutions/Infinitely Many Exactly One Solution

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