Download Solving Systems of Linear Equation in Three Variables; Finding Quadratic... I. Systems of Linear Equations in Three Variables

Math 142 — Rodriguez
Lehmann —7.6
Solving Systems of Linear Equation in Three Variables; Finding Quadratic Functions
I. Systems of Linear Equations in Three Variables
A. A linear equation in three variables is of the form Ax + By + Cz = D where A, B, C
and D are real numbers; A, B, and C are not all 0.
B. A solution to this equation is given by an ordered triple, (x,y,z).
Example: x + 2y + z = 4
Some solutions are:
(1,1,1)
(0,0,4)
(0,2,0)
The equation has an infinite number of solutions. The graph of the solutions is a plane.
A plane is a 2-dimensional surface (flat figure).
C. A system of linear equations in three variables consists of three equations of the
form Ax + By + Cz = D.
C. A solution to a system of equations consists of an ordered triple, (x,y,z), that satisfies
all three equations.
D. To solve by graphing we’d have to graph in 3 dimensions; not feasible. I will show you
some examples from the book.
II. Solving systems by eliminating variables
Idea behind method: Change the system with 3 equations & 3 variables into a system
with 2 equations & 2 variables (learned how to solve these in
Chapter 3)
Steps:
1. Choose a variable to eliminate.
2. Choose two equations and use the elimination method to eliminate the chosen
variable.
3. Choose two DIFFERENT equations and eliminate the SAME variable.
4. If the above steps are done correctly, you are left with two equations with the same
two variables. Solve this system (can use elimination or substitution). This will give
you the value of TWO of the variables.
5. Substitute the two values found into one of the original equations and solve for the
third variable.
6. Check your proposed solution in all three original equations.
*Systems can have no solution or be dependent (have many solutions). We focus on systems
that have only one solution.
Examples: Solve the system.
" x ! y + 3z = 8
$
1) # 3x + y ! 2z = !2
$2x + 4y + z = 0
%
" x +y+z=0
$
2) # x + 2y ! 3z = 5
$ 3x + 4y + 2z = !1
%
III. Application—Finding an equation of a parabola
To find equation of a parabola:
• Plug in each point into the ax 2 + bx + c = y equation. This will give you 3 equations.
• Solve the 3 by 3 system to find the values of a, b and c.
• Replace those values in y = ax 2 + bx + c .
Find an equation of a parabola that contains the given points. Use a graphing calculator to
verify that the graph of your equation contains the points.
1.
(1,1), (2, 5), (3, 15)
2.
(1,4), (2,3),
Lehmann — 7.6
(3,0)
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