Download x − 2z = −5 y − 3z = −3 2x − z = −4 x + z = 3 x + 2y − z =1

Math 1014: Precalculus with Transcendentals
Ch. 8: Systems of Equations and Inequalities
Sec. 8.2: Systems of Linear Equations in Three Variables
I.
Systems of Linear Equations in Three Variables
A. Definition
Ax + By + Cz = D , where
A, B, C and D are real numbers such that A, B and C are not all 0. A solution to a system
A linear equation in three variables is any equation of the form
of linear equations in three variables is an ordered triple of real numbers that satisfies all
equations of the system. The solution set of the system is the set of all its solutions.
B. Example
⎧ x − 2z = −5
⎪
Determine whether the ordered pair (-1,3,2) is solution of the system ⎨ y − 3z = −3 .
⎪2x − z = −4
⎩
C. Solving Linear Systems in Three Variables by Eliminating Variables
1.
2.
3.
4.
5.
Reduce the system to two equations in two variables. This is usually accomplished
by taking two different pairs of equations and using the addition method to eliminate
the same variable from each pair.
Solve the resulting system of two equations in two variables using addition or
substitution. The result is an equation in one variable that gives the value of that
variable.
Back-substitute the value found in step 2 into either of the equations in two
variables to find the value of the second variable.
Use the values of the two variables found in steps 2 and 3 to find the value of the
third variable by back-substituting into one of the original equations.
Check the proposed solution in each of the original equations.
D. Examples
1.
+z=3
⎧x
⎪
Solve ⎨ x + 2y − z = 1
⎪2x − y + z = 3
⎩
Solve the following:
2.
⎧2x + y = 2
⎪
⎨ x+ y−z = 4
⎪ 3x + 2y + z = 0
⎩
3.
⎧ x − y + 3z = 8
⎪
⎨ 3x + y − 2z = −2
⎪2x + 4y + z = 0
⎩
4.
Find the quadratic function
(3, −1), and (4, 0) .
y = ax 2 + bx + c whose graph passes through (1, 3),
5.
⎧2x + 3y + 7z = 13
⎪
Solve ⎨ 3x + 2y − 5z = −22
⎪5x + 7y − 3z = −28
⎩
6.
⎧ 7z − 3 = 2(x − 3y)
⎪
Solve ⎨5y + 3z − 7 = 4x
⎪ 4 + 5z = 3(2x − y)
⎩