1. 4/16/13 Have out: Bellwork: Solve the system of equations. A
pencil, red pen, highlighter, notebook, calculator
4/16/13
Have out:
Bellwork:
Solve the system of equations. A
2x – y + 3z = –4 B
x + 2y – 5z = 11
x + 3y – 2z = 5 C
Pick any two equations to eliminate a variable.
Let’s eliminate x.
Choose B and C
x + 2y – 5z = 11 B
x + 2y – 5z = 11 +1
–1(x + 3y – 2z) = –1(5)
x + 3y – 2z = 5
–x – 3y + 2z = –5 C +1
– y – 3z = 6 D
Label the new equation D.
total:

2. Pick another pair of equations to eliminate the x–term. A and B
2x – y + 3z = –4 B
x + 2y – 5z = 11
– y – 3z = 6 D
x + 3y – 2z = 5 C
Pick another pair of equations to eliminate the x–term.
A and B
2x – y + 3z = –4
2x – y + 3z = –4 A +1
–2x – 4y + 10z = –22
–2(x + 2y – 5z) = –2(11)
x + 2y – 5z = 11 B
–5y + 13z = –26 E +1
Label the new equation E. +1
–5(– y – 3z) = –5(6)
– y – 3z = 6 D
5y + 15z = –30
– y – 3(–2) = 6
–5y + 13z = –26 E
–5y + 13z = –26
– y + 6 = 6 +1
– y = 0
28z = –56
y = 0
x + 3y – 2z = 5
z = –2 +1 +1 +1
x + 3(0) – 2(–2) = 5
(1, 0, –2)
x = 5
x = 1 +1 +1

3. Yesterday we solved systems of equations in 3 variables.
Today we are going to look at the cases where there is no solution or infinitely many solutions.
Infinitely Many Solutions
No Solution
 planes intersect in a line
 planes intersect in the same plane
 planes have no point in common

4. Example #1: Infinitely Many Solutions
Solve the system of equations.
4x – 6y + 4z = 12 A
6x – 9y + 6z = 18 B
5x – 8y + 10z = 20 C
Eliminate x using A and B.
3(4x – 6y + 4z) = 3(12)
4x – 6y + 4z = 12 A
12x – 18y + 12z = 36
–2(6x – 9y + 6z) = –2(18)
6x – 9y + 6z = 18 B
–12x + 18y – 12z = –36
0 = 0
The equation 0 = 0 is always true. This indicates that the first two equations represent the same plane. ALWAYS check to see if this plane intersects the third plane.

5. Example #1: Infinitely Many Solutions
Solve the system of equations.
4x – 6y + 4z = 12 A
6x – 9y + 6z = 18 B
5x – 8y + 10z = 20 C
Eliminate x using A and B.
3(4x – 6y + 4z) = 3(12)
4x – 6y + 4z = 12 A
12x – 18y + 12z = 36
–2(6x – 9y + 6z) = –2(18)
6x – 9y + 6z = 18 B
–12x + 18y – 12z = –36
0 = 0
4x – 6y + 4z = 12
5(4x – 6y + 4z) = 5(12) A
20x – 30y + 20z = 60
–4(5x – 8y + 10z) = –4(20)
5x – 8y + 10z = 20 C
–20x + 32y – 40z = –80
2y – 20z = –20
Since we have an equation, the planes interest in the line.
Therefore, there are infinitely many number of solutions.

6. Example #2: No Solution
Solve the system of equations.
6x + 12y – 8z = 24 A
9x + 18y – 12z = 30 B
4x + 8y – 7z = 26 C
Eliminate x using A and B.
6x + 12y – 8z = 24
3(6x + 12y – 8z) = 3(24) A
18x + 36y – 24z = 72
–2(9x + 18y – 12z) = –2(30)
9x + 18y – 12z = 30 B
–18x – 36y + 24z = –60
0 = 12
The equation 0 = 12 is never true. Therefore, there is no solution of this system.

7. Solving Radical Equations
Solve each equation for the variable. Be sure to check for extraneous solutions.
Recall: An extraneous solution is a solution of the simplified form of an equation that does not satisfy the original equation.
For example, sometimes we had to eliminate answers when we solved logarithmic equations.
Example #1:
Step #1: Leave the radicals isolated on both sides.
Step #2: Eliminate the radicals by squaring both sides.
Step #3: Solve for x.

8. Solving Radical Equations
Example #2:
Step #1: Leave the radical isolated on the right hand side.
Step #2: Eliminate the radical by squaring both sides.
Step #3: Solve for x.
Check:
Check: √
CHECK!!!
Does NOT work.

9. Solving Radical Equations
Example #3:
Step #1: Leave the radical isolated on left hand side.
Step #2: Eliminate the radical by squaring both sides.
Step #3: Solve for x.
Check:
Check: √
CHECK!!!
Does NOT work.

10. Complete the worksheets
and the CST practice.