Example 1 consider the system defined by the

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Example 1: Consider the system defined by the following 2 equations: (i) 2 x + y = 30 and (ii) y = 4 x. We can substitute 4 x for y into the first equation to get 2 x + (4 x ) = 30, and so x = 30/6 = 5, and y = 4(5) = 20. Since the 2 equations are independent, they give a unique solution: no other pair of values than (5, 20) will satisfy both (i) and (ii). M3-1 M ATH M ODULE S olving Linear Equation Systems 3
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
M3-2 MATH MODULE 3: SOLVING LINEAR EQUATION SYSTEMS Example 2: Consider the system defined by the following 2 equations: (i) 4 x + 2 y = 60 and (ii) y = 30 – 2 x. In this case, if we substitute 30 – 2 x for y into the first equation, we end up with 0 = 0. This is because the 2 equations are dependent : if we double every term in equation (ii) and rearrange, we end up with equation (i). Hence the graphs of the 2 equations exactly coincide , and the number of solutions is infinite : any ( x , y ) pair that satisfies (i) will satisfy (ii). Example 3: Consider the system defined by the following 2 equations: (i) y = 30 – 2 x, and (ii) y = 20 – 2x. In this case, where the 2 equations are inconsistent, their graphs are parallel and therefore never intersect. Hence there is no ( x , y ) pair that satisfies both (i) and (ii). You may wish to graph these three examples, for additional practice and to ensure that you have a clear visual image of what independent, dependent, and inconsistent systems of equations look like. 2. Exercises 1. For the following sets of equations, state whether they are independent, dependent or inconsistent, and give the solutions for x and y if the equations are independent. (a) (i) 3 y = 2 x – 6 , (ii) 4 x = 12 + 6 y (b) (i) 10 = 3 x + 2 y, (ii) 20 = 3 x + 4 y (c) (i) y = 40 – x, (ii) y = x (d) (i) y = 40 – x, (ii) 3 y = x (e) (i) y = 40 – x, (ii) x = (1/3) y (f) (i) 3 y = 90 – 2 x, (ii) x = 30 – 1.5 y (g) (i) y = 30 – 2 x, (ii) y = 15 – 0.5 x (h) (i) y = 60 – 2 x, (ii) y = 30 – 0.5 x 2. Solve the following pairs of equations, and compare your results for (a) with (b), (c), and (d):
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern