http://www.ck12.org Chapter 8. Systems and Matrices
10 dollars. She could also get two onions and one basket of blueberries for 10 dollars. Write a system of equations
with variablesa,o andbrepresenting each of the three things she can buy.
- Solve the system described in # 1.
- Show that the following system is dependent.
x+y+z= 9
x+ 2 y+ 3 z= 22
2 x+ 3 y+ 4 z= 31Answers:
- Here is the system of equations:
a+o+b= 9
2 a+ 2 o= 10
2 o+b= 10- Rewrite the system usingx,yandzso thatoand 0 do not get mixed up. Include coefficients of 0 so that each
column represents one variable.
Step 1: Rewrite
1 x+ 1 y+ 1 z= 9
2 x+ 2 y+ 0 z= 10
0 x+ 2 y+ 1 z= 10Step 2: Subtract 2 times row 1 from row 2.
1 x+ 1 y+ 1 z= 9
0 x+ 0 y− 2 z=− 8
0 x+ 2 y+ 1 z= 10Step 3: Swap row 2 and row 3. Then scale row 3.
1 x+ 1 y+ 1 z= 9
0 x+ 2 y+ 1 z= 10
0 x+ 0 y+z= 4At this point you can see from the third equation thatz=4. From the second equation, 2y+ 4 =10, soy=3.
Finally you can see from the first equation thatx+ 3 + 4 =9 sox=2. Apples cost 2 dollars each, onions cost 3
dollars each and blueberries cost 4 dollars each.