Which system of equations has no solution? 3x-2y=12 6x-4y=24 x-2y=-6 3x+6y=18 4x+2y=10 6x+3y=21 2x+5y=7 3x-5y=8
The correct answer and explanation is :
To determine which system of equations has no solution, we must analyze the systems by checking if they are inconsistent. An inconsistent system of equations means that the equations represent parallel lines that never intersect, thus having no solution.
Let’s examine each system one by one.
System 1:
- ( 3x – 2y = 12 )
- ( 6x – 4y = 24 )
By inspecting the second equation, we see that it is just a multiple of the first equation. Specifically, the second equation is exactly twice the first equation (( 6x – 4y = 2 \times (3x – 2y) )). This means the two equations represent the same line, and therefore, they have infinitely many solutions. This system has a solution.
System 2:
- ( x – 2y = -6 )
- ( 3x + 6y = 18 )
We can solve this system by substitution or elimination. From the first equation, solve for ( x ):
[ x = 2y – 6 ]
Substitute this expression for ( x ) into the second equation:
[
3(2y – 6) + 6y = 18 \implies 6y – 18 + 6y = 18 \implies 12y = 36 \implies y = 3
]
Substitute ( y = 3 ) back into ( x = 2y – 6 ):
[ x = 2(3) – 6 = 6 – 6 = 0 ]
Thus, the solution is ( x = 0 ), ( y = 3 ). This system has a solution.
System 3:
- ( 4x + 2y = 10 )
- ( 6x + 3y = 21 )
To check for consistency, we can multiply the first equation by 3 and the second equation by 2:
[
3(4x + 2y) = 3(10) \implies 12x + 6y = 30
]
[
2(6x + 3y) = 2(21) \implies 12x + 6y = 42
]
This results in:
[
12x + 6y = 30 \quad \text{and} \quad 12x + 6y = 42
]
These are contradictory equations because they suggest that the same expression equals two different values. This system has no solution.
System 4:
- ( 2x + 5y = 7 )
- ( 3x – 5y = 8 )
We can solve this system using the elimination method. Add the two equations to eliminate ( y ):
[
(2x + 5y) + (3x – 5y) = 7 + 8 \implies 5x = 15 \implies x = 3
]
Substitute ( x = 3 ) into the first equation:
[
2(3) + 5y = 7 \implies 6 + 5y = 7 \implies 5y = 1 \implies y = \frac{1}{5}
]
Thus, the solution is ( x = 3 ), ( y = \frac{1}{5} ). This system has a solution.
Conclusion:
The system with no solution is System 3:
- ( 4x + 2y = 10 )
- ( 6x + 3y = 21 )
This system is inconsistent because the equations represent parallel lines that never intersect.