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Math Problems
Precalculus
Determine the number of solutions to a system of equations in three variables
How many solutions does the system of equations below have?
\newline
2
x
+
y
+
2
z
=
11
2x + y + 2z = 11
2
x
+
y
+
2
z
=
11
\newline
−
2
x
−
3
y
+
3
z
=
19
-2x - 3y + 3z = 19
−
2
x
−
3
y
+
3
z
=
19
\newline
2
x
+
3
y
−
3
z
=
15
2x + 3y - 3z = 15
2
x
+
3
y
−
3
z
=
15
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
3
x
−
3
y
+
3
z
=
−
18
3x - 3y + 3z = -18
3
x
−
3
y
+
3
z
=
−
18
\newline
x
−
y
+
z
=
−
6
x - y + z = -6
x
−
y
+
z
=
−
6
\newline
−
2
x
+
2
y
−
2
z
=
12
-2x + 2y - 2z = 12
−
2
x
+
2
y
−
2
z
=
12
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
−
3
x
−
3
y
+
3
z
=
12
-3x - 3y + 3z = 12
−
3
x
−
3
y
+
3
z
=
12
\newline
2
x
+
2
y
−
2
z
=
−
8
2x + 2y - 2z = -8
2
x
+
2
y
−
2
z
=
−
8
\newline
x
+
y
−
z
=
−
4
x + y - z = -4
x
+
y
−
z
=
−
4
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
3
x
+
y
+
3
z
=
−
5
3x + y + 3z = -5
3
x
+
y
+
3
z
=
−
5
\newline
2
x
−
3
y
−
3
z
=
16
2x - 3y - 3z = 16
2
x
−
3
y
−
3
z
=
16
\newline
3
x
+
2
y
+
z
=
−
17
3x + 2y + z = -17
3
x
+
2
y
+
z
=
−
17
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
−
x
−
2
y
−
z
=
14
-x - 2y - z = 14
−
x
−
2
y
−
z
=
14
\newline
−
2
x
+
y
+
3
z
=
18
-2x + y + 3z = 18
−
2
x
+
y
+
3
z
=
18
\newline
2
x
−
y
−
3
z
=
−
17
2x - y - 3z = -17
2
x
−
y
−
3
z
=
−
17
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
2
x
+
2
y
−
2
z
=
−
10
2x + 2y - 2z = -10
2
x
+
2
y
−
2
z
=
−
10
\newline
−
x
−
2
y
−
3
z
=
15
-x - 2y - 3z = 15
−
x
−
2
y
−
3
z
=
15
\newline
2
x
+
3
y
+
2
z
=
−
20
2x + 3y + 2z = -20
2
x
+
3
y
+
2
z
=
−
20
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
x
+
3
y
−
z
=
−
5
x + 3y - z = -5
x
+
3
y
−
z
=
−
5
\newline
−
x
−
3
y
+
z
=
5
-x - 3y + z = 5
−
x
−
3
y
+
z
=
5
\newline
2
x
+
3
y
+
z
=
17
2x + 3y + z = 17
2
x
+
3
y
+
z
=
17
\newline
Choices:
\newline
(A) no solution
\newline
(B) one solution
\newline
(C) infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
2
x
−
2
y
+
2
z
=
−
10
2x - 2y + 2z = -10
2
x
−
2
y
+
2
z
=
−
10
\newline
−
2
x
+
y
−
3
z
=
6
-2x + y - 3z = 6
−
2
x
+
y
−
3
z
=
6
\newline
−
x
−
y
−
2
z
=
−
6
-x - y - 2z = -6
−
x
−
y
−
2
z
=
−
6
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
−
3
x
+
y
−
z
=
−
10
-3x + y - z = -10
−
3
x
+
y
−
z
=
−
10
\newline
3
x
−
3
y
+
2
z
=
−
18
3x - 3y + 2z = -18
3
x
−
3
y
+
2
z
=
−
18
\newline
3
x
+
3
y
−
z
=
−
14
3x + 3y - z = -14
3
x
+
3
y
−
z
=
−
14
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
3
x
+
3
y
+
2
z
=
15
3x + 3y + 2z = 15
3
x
+
3
y
+
2
z
=
15
\newline
−
2
x
−
2
y
+
z
=
11
-2x - 2y + z = 11
−
2
x
−
2
y
+
z
=
11
\newline
2
x
−
y
+
z
=
4
2x - y + z = 4
2
x
−
y
+
z
=
4
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
3
x
−
y
−
3
z
=
9
3x - y - 3z = 9
3
x
−
y
−
3
z
=
9
\newline
−
x
+
y
+
2
z
=
−
11
-x + y + 2z = -11
−
x
+
y
+
2
z
=
−
11
\newline
−
2
x
+
2
y
+
z
=
−
10
-2x + 2y + z = -10
−
2
x
+
2
y
+
z
=
−
10
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
2
x
−
3
y
+
z
=
9
2x - 3y + z = 9
2
x
−
3
y
+
z
=
9
\newline
−
2
x
−
3
y
+
2
z
=
−
18
-2x - 3y + 2z = -18
−
2
x
−
3
y
+
2
z
=
−
18
\newline
−
2
x
+
2
y
+
2
z
=
12
-2x + 2y + 2z = 12
−
2
x
+
2
y
+
2
z
=
12
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
−
3
x
−
y
+
z
=
−
16
-3x - y + z = -16
−
3
x
−
y
+
z
=
−
16
\newline
2
x
+
y
+
z
=
5
2x + y + z = 5
2
x
+
y
+
z
=
5
\newline
−
x
−
3
y
+
2
z
=
10
-x - 3y + 2z = 10
−
x
−
3
y
+
2
z
=
10
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
−
3
x
−
3
y
+
3
z
=
−
2
-3x - 3y + 3z = -2
−
3
x
−
3
y
+
3
z
=
−
2
\newline
x
+
y
−
z
=
−
1
x + y - z = -1
x
+
y
−
z
=
−
1
\newline
−
x
−
y
+
z
=
−
7
-x - y + z = -7
−
x
−
y
+
z
=
−
7
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
3
x
+
y
−
z
=
−
5
3x + y - z = -5
3
x
+
y
−
z
=
−
5
\newline
−
2
x
−
2
y
+
z
=
0
-2x - 2y + z = 0
−
2
x
−
2
y
+
z
=
0
\newline
3
x
+
2
y
−
3
z
=
8
3x + 2y - 3z = 8
3
x
+
2
y
−
3
z
=
8
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
x
+
y
+
2
z
=
19
x + y + 2z = 19
x
+
y
+
2
z
=
19
\newline
−
x
−
y
−
2
z
=
−
19
-x - y - 2z = -19
−
x
−
y
−
2
z
=
−
19
\newline
2
x
−
y
+
z
=
1
2x - y + z = 1
2
x
−
y
+
z
=
1
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
3
x
+
y
+
2
z
=
−
15
3x + y + 2z = -15
3
x
+
y
+
2
z
=
−
15
\newline
−
3
x
−
y
−
2
z
=
15
-3x - y - 2z = 15
−
3
x
−
y
−
2
z
=
15
\newline
−
2
x
−
2
y
+
z
=
12
-2x - 2y + z = 12
−
2
x
−
2
y
+
z
=
12
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
−
2
x
+
2
y
−
2
z
=
10
-2x + 2y - 2z = 10
−
2
x
+
2
y
−
2
z
=
10
\newline
−
x
+
y
−
z
=
5
-x + y - z = 5
−
x
+
y
−
z
=
5
\newline
2
x
−
2
y
+
2
z
=
−
10
2x - 2y + 2z = -10
2
x
−
2
y
+
2
z
=
−
10
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
\newline
Get tutor help
How many solutions does the system of equations below have?
\newline
2
x
−
2
y
−
z
=
3
2x - 2y - z = 3
2
x
−
2
y
−
z
=
3
\newline
−
x
−
y
+
z
=
−
12
-x - y + z = -12
−
x
−
y
+
z
=
−
12
\newline
−
3
x
+
y
+
3
z
=
−
12
-3x + y + 3z = -12
−
3
x
+
y
+
3
z
=
−
12
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
−
x
+
3
y
−
z
=
17
-x + 3y - z = 17
−
x
+
3
y
−
z
=
17
\newline
3
x
−
y
−
z
=
−
20
3x - y - z = -20
3
x
−
y
−
z
=
−
20
\newline
x
−
3
y
+
z
=
−
17
x - 3y + z = -17
x
−
3
y
+
z
=
−
17
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
−
x
+
2
y
+
3
z
=
−
10
-x + 2y + 3z = -10
−
x
+
2
y
+
3
z
=
−
10
\newline
−
3
x
−
2
y
−
2
z
=
−
4
-3x - 2y - 2z = -4
−
3
x
−
2
y
−
2
z
=
−
4
\newline
−
3
x
+
2
y
−
z
=
10
-3x + 2y - z = 10
−
3
x
+
2
y
−
z
=
10
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
−
3
x
+
3
y
+
z
=
−
6
-3x + 3y + z = -6
−
3
x
+
3
y
+
z
=
−
6
\newline
3
x
−
3
y
−
z
=
6
3x - 3y - z = 6
3
x
−
3
y
−
z
=
6
\newline
−
2
x
+
2
y
−
3
z
=
11
-2x + 2y - 3z = 11
−
2
x
+
2
y
−
3
z
=
11
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
Google Classroom
\newline
Answer two questions about Systems
A
A
A
and
B
B
B
:
\newline
System
A
A
A
\newline
{
x
+
3
y
=
−
9
2
x
+
y
=
4
{
3
x
+
4
y
=
−
9
2
x
+
y
=
4
\left\{\begin{array}{l}x+3 y=-9 \\ 2 x+y=4\end{array} \quad\left\{\begin{array}{l}3 x+4 y=-9 \\ 2 x+y=4\end{array}\right.\right.
{
x
+
3
y
=
−
9
2
x
+
y
=
4
{
3
x
+
4
y
=
−
9
2
x
+
y
=
4
\newline
1
1
1
) How can we get System
B
B
B
from System
A
A
A
?
\newline
Choese
1
1
1
answer:
\newline
(A) Replace one equation with the sum/difference of both equations
\newline
Replace only the left-hand side of one equation with the sum/difference of the left-hand sides of both equations
\newline
(C) Replace one equation with a multiple of itself
\newline
(D) Replace one equation with a multiple of the other equation
\newline
2
2
2
) Based on the previous answer, are the systems equivalent? In other words, do they have the same solution?
\newline
Choese
1
1
1
answer:
\newline
(A) Yes
\newline
(D) No
Get tutor help
How many solutions does the system of equations below have?
\newline
y
=
10
x
+
1
y = 10x + 1
y
=
10
x
+
1
\newline
y
=
10
x
+
1
y = 10x + 1
y
=
10
x
+
1
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
y
=
9
x
+
5
y = 9x + 5
y
=
9
x
+
5
\newline
y
=
9
x
−
3
4
y = 9x - \frac{3}{4}
y
=
9
x
−
4
3
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
y
=
−
2
7
x
−
1
y = -\frac{2}{7}x - 1
y
=
−
7
2
x
−
1
\newline
y
=
−
2
7
x
−
1
y = -\frac{2}{7}x - 1
y
=
−
7
2
x
−
1
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
How many solutions does the system of equations below have?
\newline
y
=
2
3
x
+
6
y = \frac{2}{3}x + 6
y
=
3
2
x
+
6
\newline
y
=
2
3
x
+
6
y = \frac{2}{3}x + 6
y
=
3
2
x
+
6
\newline
Choices:
\newline
(A)no solution
\newline
(B)one solution
\newline
(C)infinitely many solutions
Get tutor help
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
\newline
2
x
+
3
y
=
4
4
x
+
6
y
=
8
\begin{array}{l} 2 x+3 y=4 \\ 4 x+6 y=8 \end{array}
2
x
+
3
y
=
4
4
x
+
6
y
=
8
\newline
No Solutions
\newline
One Solution
\newline
Infinitely Many Solutions
Get tutor help
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
\newline
x
+
3
y
=
9
−
2
x
−
6
y
=
−
18
\begin{aligned} x+3 y & =9 \\ -2 x-6 y & =-18 \end{aligned}
x
+
3
y
−
2
x
−
6
y
=
9
=
−
18
\newline
No Solutions
\newline
Infinitely Many Solutions
\newline
One Solution
Get tutor help
