Solved Examples and Worksheet for Finding Number of Solutions of System of Linear Equations

Q1Using the substitution method, find the number of solutions the given linear system has.
12x + 4y = -21
-3x - y = 5

A. Infinitely many solutions
B. No solution
C. Exactly one solution
D. Cannot be determined

Solutions
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Step: 1
12x + 4y = -21
  [Equation 1.]
Step: 2
-3x - y = 5
  [Equation 2.]
Step: 3
y = -3x - 5
  [Revise equation 2.]
Step: 4
12x + 4(-3x - 5) = -21
  [Substitute -3x - 5 for y in equation 1.]
Step: 5
20 ≠ 21
  [Combine like terms.]
Step: 6
The variables are eliminated and the resulting statement is false.
Step: 7
So, the linear system has no solution.
Correct Answer is :   No solution
Q2Find the number of solutions the linear system has.
5x + y = 3
5x + y = 2

A. exactly one solution
B. infinitely many solutions
C. no solution
D. exactly two solutions

Solutions
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Step: 1
5x + y = 3
  [Equation 1.]
Step: 2
5x + y = 2
  [Equation 2.]
Step: 3
y = - 5x + 2
  [From Equation 2.]
Step: 4
5x + (- 5x + 2) = 3
  [Substitute y = - 5x + 2 in Equation 1.]
Step: 5
2 = 3
  [Combine like terms.]
Step: 6
The variables are eliminated and the resulting statement is false.
Step: 7
So, the linear system has no solution.
Correct Answer is :   no solution
Q3Find the number of solutions the linear system has.
6x + y = 4
6x + y = 3
A. exactly two solutions
B. no solution
C. exactly one solution
D. infinitely many solutions

Solutions
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Step: 1
6x + y = 4
  [Equation 1.]
Step: 2
6x + y = 3
  [Equation 2.]
Step: 3
y = - 6x + 3
  [From Equation 2.]
Step: 4
6x + (- 6x + 3) = 4
  [Substitute y = - 6x + 3 in Equation 1.]
Step: 5
3 = 4
  [Combine like terms.]
Step: 6
The variables are eliminated and the resulting statement is false.
Step: 7
So, the linear system has no solution.
Correct Answer is :   no solution
Q4Use the substitution method to find the number of solutions the linear system has.
9x + 3y = - 7
- 3x - y = 2
A. infinitely many solutions
B. no solution
C. exactly one solution
D. exactly two solutions

Solutions
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Step: 1
9x + 3y = - 7
  [Equation 1.]
Step: 2
- 3x - y = 2
  [Equation 2.]
Step: 3
y = - 3x - 2
  [Revise equation 2.]
Step: 4
9x + 3(- 3x - 2) = - 7
  [Replace y with - 3x - 2 in Equation 1.]
Step: 5
9x - 9x - 6 = - 7
  
Step: 6
6 = 7
  
Step: 7
The variables are eliminated and the resulting statement is false.
Step: 8
So, the linear system has no solution.
Correct Answer is :   no solution
Q5Find the number of solutions the linear system has.
4x + y = 29
5x - 6y = 0
A. two solutions
B. no solution
C. exactly one solution
D. infinitely many solutions

Solutions
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Step: 1
4x + y = 29
  [Equation 1.]
Step: 2
5x - 6y = 0
  [Equation 2.]
Step: 3
y = - 4x + 29
  [Rearrange Equation 1.]
Step: 4
5x - 6(- 4x + 29) = 0
  [Replace y with - 4x + 29 in Equation 2.]
Step: 5
5x + 24x - 174 = 0
  [Use distributive property.]
Step: 6
29x - 174 = 0
  [Combine like terms.]
Step: 7
29x = 174
  [Add 174 to both sides.]
Step: 8
x = 6
  [Solve for x.]
Step: 9
y = - 4(6) + 29
  [Multiply.]
Step: 10
y = 5
  [Solve for y.]
Step: 11
So, the ordered pair (6, 5) is the solution for the linear system.
Step: 12
So, the linear system has exactly one solution.
Correct Answer is :   exactly one solution
Q6Find the number of solutions the linear system has.
8x + y = 6
8x + y = 7

A. Two solutions
B. Zero solutions
C. Exactly one solution
D. Infinitely many solutions

Solutions
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Step: 1
8x + y = 6
  [Equation 1.]
Step: 2
8x + y = 7
  [Equation 2.]
Step: 3
y = - 8x + 7
  [From Equation 2.]
Step: 4
8x + (- 8x + 7) = 6
  [Substitute the values.]
Step: 5
7 = 6
  [Group the like terms.]
Step: 6
The variables are eliminated and the resulting statement, 7 = 6 is false.
Step: 7
So, the linear system has no solution.
Correct Answer is :   Zero solutions
Q7Find the number of solutions the linear system has.
x - 3y = 7
x - 3y = 4
A. no solution
B. exactly two solutions
C. infinitely many solutions
D. exactly one solution

Solutions
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Step: 1
x - 3y = 7
  [Equation 1.]
Step: 2
x - 3y = 4
  [Equation 2.]
Step: 3
x = 3y + 4
  [From Equation 2.]
Step: 4
(3y + 4) - 3y = 7
  [Replace x = 3y + 4 in Equation 1.]
Step: 5
4 = 7
  [Combine like terms.]
Step: 6
The variables are eliminated and the resulting statement is false.
Step: 7
So, the linear system has no solution.
Correct Answer is :   no solution
Q8Find the number of solutions the linear system has.
4 x + y = 2
4x + y = 1

A. no solution
B. exactly two solutions
C. exactly one solution
D. infinitely many solutions

Solutions
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Step: 1
4x + y = 2
  [Equation 1.]
Step: 2
4x + y = 1
  [Equation 2.]
Step: 3
y = - 4x + 1
  [From Equation 2.]
Step: 4
4x + (- 4x + 1) = 2
  [Substitute y = - 4x + 1 in equation]
Step: 5
1 = 2
  [Combine like terms.]
Step: 6
The variables are eliminated and the resulting statement is false.
Step: 7
So, the linear system has no solution.
Correct Answer is :   no solution
Q9Use the substitution method to find the number of solutions the linear system has.
10x + 5y = - 11
- 2x - y = 2
A. exactly one solution
B. no solution
C. exactly two solutions
D. infinitely many solutions

Solutions
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Step: 1
10x + 5y = - 11
  [Equation 1.]
Step: 2
- 2x - y = 2
  [Equation 2.]
Step: 3
y = - 2x - 2
  [Revise equation 2.]
Step: 4
10x + 5(- 2x - 2) = - 11
  [Replace y with - 2x - 2 in Equation 1.]
Step: 5
10x - 10x - 10 = - 11
Step: 6
10 = 11
Step: 7
The variables are eliminated and the resulting statement is false.
Step: 8
So, the linear system has no solution.
Correct Answer is :   no solution
Q10Use linear combinations to find the number of solutions the linear system has.
- 8x + y = - 4
160x - 20y = 80

A. infinitely many solutions
B. two solutions
C. exactly one solution
D. no solution

Solutions
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Step: 1
- 8x + y = - 4
  [Equation 1.]
Step: 2
160x - 20y = 80
  [Equation 2.]
Step: 3
160x - 20y = 80
  [Multiply Equation 1 by -20.]
Step: 4
The two equations are identical.
Step: 5
Any solution of 160x - 20y = 80 is also a solution of the system.
Step: 6
So, the linear system has infinitely many solutions.
Correct Answer is :   infinitely many solutions

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