Name:________________________
May 1, 2014 (A Block) or May 5, 2014 (E Block)
Algebra 2
How Many Solutions to a System?
1. Suppose that each of these matrices is the end result of an attempt to solve a system using
the matrix elimination method or the rref command on the calculator. Tell whether the
system has no solutions, one solution, or infinitely many solutions. If there is one solution,
also write the solution.
é1 0 2ù
a. ê
ú
ë0 1 4 û
é1 - 1 0ù
b. ê
ú
ë0 0 1 û
é1 3 2ù
c. ê
ú
ë0 0 0 û
é1 0 5ù
d. ê
ú
ë0 1 0 û
é1 0 3ù
ê
ú
e. ê0 1 4ú
êë0 0 0úû
f.
é1 0 0ù
ê0 1 0 ú
ê
ú
êë0 0 1úû
é1 0 0 2ù
g. ê0 1 0 1 ú
ê
ú
êë0 0 0 0úû
é1 0 0 2ù
h. ê0 1 0 1 ú
ê
ú
êë0 0 1 0úû
é1 0 0 2ù
ê0 1 0 1 ú
ê
ú
êë0 0 0 1 úû
j.
é1 0 0 0ù
ê0 1 0 0 ú
ê
ú
êë0 0 1 0úû
l.
é1 0 0 0 1ù
ê0 1 0 0 3 ú
ê
ú
êë0 0 1 0 2úû
i.
é1 0 4 1ù
k. ê
ú
ë0 1 2 3û
Name:________________________
May 1, 2014 (A Block) or May 5, 2014 (E Block)
Algebra 2
2. An elementary school store sells pencils and erasers. There’s supposed to be a fixed price for
pencils and a fixed price for erasers.
Evan had to pay $0.70 to buy 2 pencils and 1 eraser.
Renee had to pay $1.05 to buy 3 pencils and 2 erasers.
Nathan had to pay $1.40 to buy 5 pencils and 1 eraser.
Prove that at least one of the children was charged an incorrect amount.
Hint: Let x and y stand for the prices. Write a system of 3 equations. Use rref on your
calculator to show that the system has no solutions.
Name:________________________
May 1, 2014 (A Block) or May 5, 2014 (E Block)
Algebra 2
3. Continuation from problem 2: The teacher overseeing the school store determined that Evan
had been charged the wrong price. She corrected the amount that Evan had to pay, and made
sure that everyone paid the correct prices.
Evan had to pay $0.65 to buy 2 pencils and 1 eraser.
Renee had to pay $1.05 to buy 3 pencils and 2 erasers.
Nathan had to pay $1.40 to buy 5 pencils and 1 eraser.
Find the price of a pencil and the price of an eraser.
Name:________________________
May 1, 2014 (A Block) or May 5, 2014 (E Block)
Algebra 2
4. Solve each system using the matrix elimination method (no calculator!). If there is one
solution, give the solution. Otherwise, state that there are no solutions or infinitely many
solutions.
a.
3x – 6y = 15
–2x + 4y = 10
b.
3x – 6y = 15
–2x + 4y = –10
c.
3x – 6y = 15
–2x – 4y = 10
Name:________________________
May 1, 2014 (A Block) or May 5, 2014 (E Block)
Algebra 2
5. Solve these linear systems using the rref command on your calculator. If it turns out that a
system has no solutions, or infinitely many solutions, say so.
a. x + y + z = 5
x + 2y + 3z = 6
–x
+z=4
b. x + y + z = 5
x + 2y + 3z = 6
–x
+ z = –4
c. 3x – 6y = 15
–2x + 4y = 10
d. 3x – 6y = 15
–2x + 4y = –10
e. w + 2x + 3y + 4z = 5
6w + 7x + 8y + 9z = 10
11w + 12x + 13y + 14z = 15
16w + 17x + 18y + 19z = 20
Name:________________________
May 1, 2014 (A Block) or May 5, 2014 (E Block)
Algebra 2
6. Here are amounts that were charged to three families for tickets at a movie theater.
$37.75 for 2 adults and 3 children
$33.75 for 1 adult and 4 children
$35.80 for 2 adults and 2 children
Show that at least one family must have been incorrectly charged. Hint: Set up a linear
system and show that it has no solution. (You may use a calculator)