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Systems of Linear Equations

This comprehensive guide explores the role of determinants in systems of linear equations, helping you understand the implications of various determinant values. Discover how to determine if a system has no solutions, infinitely many solutions, or exactly one solution. Learn through practical examples and exercises that illustrate the concepts of parallel lines, the same line, and intersections in graphs of linear equations. Enhance your problem-solving skills with practice problems that reinforce your understanding of determinants and their significance in linear algebra.

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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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