Chapter 7 Sec 3a

Multivariable Linear Systems

2 of 13

Pre Calculus Ch 7.3a

Essential Question

How do you solve systems of equations with more than two

variables?

Key Vocabulary:Dependent/ Independent

3 of 13

Pre Calculus Ch 7.3aRow-Echelon Form and Back Substitution

System of Three Linear Equations in Three Variables

Equivalent System in Row-Echelon Form

This 2nd system is row-echelon, which means it has a stair step pattern with leading coefficients of 1.

17552

23

932

zyx

zyx

zyx

2

74

932

z

zy

zyx

4 of 13

Pre Calculus Ch 7.3aExample 1: Use Back-substitution in Row-Echelon form

Solve the system of linear equations.

From Equationv3, you know z. To solve for y, substitute z = 2 in Equation 2.

y + 4(2) = 7 … y = –1

Finally substitute y = –1 and z = 2 into Equation 1,

x – 2(–1) + 3(2) = 9 … x = 1

We now can write our solution as an ordered triple (1, –1, 2)

2

74

932

z

zy

zyx Equation 1

Equation 2

Equation 3

5 of 13

Pre Calculus Ch 7.3aGaussian Elimination

• Two system of equations are equivalent if they have the same solution.

• To solve a system not in row-echelon form, first convert it to a equivalent system that is in row-echelon form by using one or more of the elementary row operations.

• This process is called Gaussian elimination, after Carl Friedrich Gauss (1777 – 1855).

Elementary Row Operations for Systems of Equations

1. Interchange two equations.

2. Multiply one of the equations by a non-zero constant.

3. Add a multiple of one equation to another

6 of 13

Pre Calculus Ch 7.3aExample 2: Use Gaussian Elimination

17552

23

932

zyx

zyx

zyx

17552

74

932

zyx

zy

zyx

Equation 1

Equation 2

Equation 3

Adding the Equations 1 & 2 give new Equation 2

1

74

932

zy

zy

zyx Adding –2 x Eq. 1 to Eq 3 give new Equation 3

63

74

932

z

zy

zyx

2

74

932

z

zy

zyx

Adding Eq. 2 & 3 give new Equation 3

Multiply Equation 3 by 1/3 gives…

Solve the system of linear equations.

Because the leading coefficient of the first equation is 1, begin by eliminating the other x terms from the first column.

Now use back substitution to solve for (1, –1, 2).

7 of 13

Pre Calculus Ch 7.3aExample 3: Inconsistent System

Solve the system of linear equations.

Solution.

Equation 1

Equation 2

Equation 3

132

222

13

zyx

zyx

zyx

Adding –2 x Eq. 1 to Eq 2 give new Equation 2

132

045

13

zyx

zy

zyx

245

045

13

zy

zy

zyx Adding –1 x Eq. 1 to Eq 3 give new Equation 3

20

045

13

zy

zyxAdding –1 x Eq. 2 to Eq 3 give new Equation 3

Because 0 = –2 is a false statement, you can conclude that this system is inconsistent and has no solution.

8 of 13

Pre Calculus Ch 7.3aNumbers of Solutions

• A system of linear equations is called consistent if it has at least one solution.

• A consistent system with exactly one solution is independent.

• A consistent system with infinite many solutions is dependent.

• A system of linear equations is called inconsistent if it has no solution.

Number of solutions of a Linear SystemFor a system of linear equations, exactly one of the following is true.

1. There is exactly one solution..

2. There are infinite many solutions. (true statement)

3. There is no solution. (false statement as in previous example)

9 of 13

Pre Calculus Ch 7.3aExample 4: Infinite Solutions

Solve the system of linear equations.

Solution.

Equation 1

Equation 2

Equation 3

Adding Eq. 1 to Eq 3 give new Equation 3

Because 0 = 0 is a true statement, you have infinite many solutions.

1 2

0

13

yx

zy

zyx

033

0

13

zy

zy

zyx

00

0

13

zy

zyx

0

0

13

zy

zy

zyx

0

13

zy

zyx

10 of 13

Pre Calculus Ch 7.3aExample 4: Infinite Solutions

We now have the equivalent system.

Solve last equation in terms of z to obtain y = z. Back substituting for y produces x = 2z – 1.

Finally let z = a, where a is a real number, we get.

x = 2a – 1, y = a, and z = a.

So, every ordered triple of the form (2a – 1, a, a) is a solution of the system.

0

13

zy

zyx

11 of 13

Pre Calculus Ch 7.3a

Systems of Linear Equations in Three Variables

12 of 13

Pre Calculus Ch 7.3a

Essential Question

How do you solve systems of equations with more than

two variables?

13 of 13

Pre Calculus Ch 7.3a

Daily Assignment

• Chapter 7 Section 3a• Text Book

• Pg 505 – 506 • #1 – 29 Odd

• Show all work for credit.