Simultaneous Equations

Graphing Calculator

Grade: Years 9-10
KLA: Mathematics
Topic: Linear and quadratic equations.
   
Overview:  

In this activity, students solve two simultaneous equations, one being quadratic and the other being linear using the graphing calculator in Microsoft Student 2006.

 
Key Learning Area Outcomes:
  • Mathematics Outcomes:
    • Understand what a quadratic and linear equation is.

    • Understand how to solve a quadratic and linear equation simultaneously.

Application:
  • Microsoft Student 2006
Teacher Notes:
  1. Introduce the activity by explaining that quadratic equations are different from linear equations by the fact that they include a polynomial in the equation. Explain that solving a quadratic equation means that we are finding the values for x that make y equal 0.

  2. Have the students solve simple quadratic equations and simple linear equations separately.

  3. Once students have attempted to answer the questions, have them check their work using the graphing calculator using Microsoft Student 2006.


Graphing Calculator - Microsoft Student 2006

Student Activity:
  1. The type of simultaneous equations we shall be dealing with are of the form ax2 + bx + c = y and y = mx + c where a, b, c and m are constant variables and a not equal to 0 for ax2 + bx + c = y. Where will the solutions of such equations be located on a graph?

  2. Let's warm up by solving the following equations:

    i)    x2 - 5x + 6 = y, y = x - 1

    ii)   2x2 - 3x - 5 = y, (2x + 2)/4 = y

    iii)  2x + y = 4, 2x2 + 5x - 3 = y

    iv)   a2 - 3a + 5 = b, a + b = 5

    Hint! y = x2 - 5x + 6 and y = x - 1 can be written as x2 - 5x + 6 = x - 1 since they are both equal to y and as such they can be equated together (think of substitution method). This idea is very useful when solving such equations, since graphing calculator can not allow you to type two equations in the form of that in (i) to (iii) above.

    Solutions

    i)    x = 4.41421 and y = 3.541421

    or x = 1.58579 and y = 0. 585786

    ii)   x = 4.63314 and y = 2.56657

    or x= -1.13314 and -0.31657

    iii) x = 0.811738 and y = 2.37652

    x = -4.31174 and 12.6235

    iv) a = 2 and b = 3

    a = 0 and b = 5

     

  3. Solving two quadratic equations.

    The solution of these two will also be given by the intersection of the two curves. Try and solve the following simultaneous equations.

    i) x2 - 5x + 6 = y , 2x2 + 3x - 4 = y

    ii) 4x2 - x + 3 = y , y = x2 - 5x + 10

    iii) a2 + 5b + 6 = b, b = 2a2 - 3a - 7

    iv) a2 -3b + 5 = 3a, 5b - 3a2 + 8 = 4

Solutions:

i) x= -9.09902 and y = 134.287  or x= 1.09902 and y = 1.71275

ii) x = 1 and y = 6 or x = -2.33333 and y = 27.1111

iii) a = -1.38516 and b = 0.992858 a = 9.38516 and b = 141.007

iv) a = 1.6979 and b = 0.929722 a = -5.4479 and b = 17.0078

Extension:
You may like to extend this activity by having students solving two simultaneous equations: one linear and the other non-linear. Like in the previous cases above, the solution of these equations will be given by the intersection of the two graphs if they intersect. What do you think will happen when the two have no solutions?  Using your previous knowledge try to solve the following simultaneous equations.

i) x + y = 9 and xy = 8 [make y the subject of each equation].

ii) xy = 2 and y = 3x

iii) x(y - 1) = 4 and 6 - y = 2x

iv) 2ab + 4 = 0 and a + b = 7

v) k + 2/h = 4 and 2k + h = -4

Solutions:

i) x = 8 and y = 1 or x = 1 and y = 8

ii) x = 0.816497 and y = -2.44949 or x = -0.816497 and y = -2.44949

iii) x = -3.13746 and y = -0.274917 or x = 0.637459 and y = 7.27492

iv) a = -0.274917 and b= 7.27492 or x = 7.27492 and b = -0.274917

v) k = -12.3246 and h = 4.16228 or k = 0.324555 and h = -2.16228

* Lesson courtesy of Sonny Mooketski

 
Assessment:
Students answers to the above questions should indicate their success at this activity.