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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
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
−
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
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
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
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
