Elimination Method Quiz: Elimination Impossible Statement

Add the equations to eliminate y: (3x+2y) + (5x-2y) = 14 + 6, giving 8x = 20, so x = 5/2. Substitute into 3x+2y=14: 15/2 + 2y = 14, so 2y = 13/2, y = 13/4. Check: 3(5/2)+2(13/4) = 15/2+13/2 = 14 ✓ and 5(5/2)-2(13/4) = 25/2-13/2 = 6 ✓. The solution is (5/2, 13/4).

Double the first equation: 2x - 4y = 8. Subtract from 2x - 4y = 9: 0 = 9 - 8 = 1, a false statement. The second equation is almost twice the first but with an inconsistent constant, confirming the lines are parallel and never intersect. The system has no solution. Options A and C incorrectly assume a unique solution. Option D would require the doubled equation to match the second exactly.

The answer is False. The statement 12=12 is a true identity, not a contradiction. When elimination produces a true numerical statement after all variables cancel, the two equations are dependent — they represent the same line. This means every point on the line satisfies both equations, giving infinitely many solutions, not a unique one. A unique solution requires finding specific numerical values for each variable.

A contradiction after elimination means the two equations represent parallel lines. Parallel lines have equal slopes confirming B, and since they are distinct parallel lines their y-intercepts must differ confirming C. Parallel lines with different intercepts never intersect confirming A. Since there is no intersection point the system has no solution confirming D. All four statements are necessarily true whenever elimination yields a false numerical statement.

Add the equations: (2x + y) + (5x - y) = 9 + 6, giving 7x = 15, so x = 15/7. Option A gives 3, which would require 7x=21. Option C gives 2, requiring 7x=14. Option D gives 5/3, requiring 7x=35/3. Only 15/7 correctly results from adding the two equations.

Add equations to eliminate y: (4x-3y) + (2x+3y) = 13 + 1, giving 6x = 14, so x = 7/3. Substitute into 2x+3y=1: 14/3 + 3y = 1, so 3y = 1 - 14/3 = -11/3, y = -11/9. Check: 4(7/3)-3(-11/9) = 28/3+11/3 = 39/3 = 13 ✓. The solution is (7/3, -11/9).

Multiply the first equation by 2: 6x + 10y = 2. Subtract from 6x + 10y = 3: (6x+10y) - (6x+10y) = 3 - 2, giving 0 = 1. This is a false statement. The second equation is almost twice the first but with an inconsistent constant, making the lines parallel with no intersection. Options B, C, and D all incorrectly assume a solution exists.

The answer is True. Proportional coefficients mean the left-hand sides of both equations are scalar multiples of each other. When scaled to match and then subtracted, all variable terms cancel. If the constants are not in the same proportion, the result is a nonzero number equal to zero, a false statement confirming the lines are parallel with no solution.

Option B: double first gives 2x-4y=2, subtract from 2x-4y=5 gives 0=3, contradiction. Option D: double first gives 8x-2y=16, subtract from 8x-2y=17 gives 0=1, contradiction. Option A: double first gives 2x+2y=8, identical to second — an identity with infinite solutions. Option C: double first gives 4x+2y=10, identical to second — also an identity with infinite solutions.

The statement 0 = -12 is a false numerical equation with no variables remaining. When elimination reduces a system to a false constant statement, the equations represent parallel lines that never intersect. The system is inconsistent and has no solution. Options A and B incorrectly suggest solutions exist. Option C attempts to extract variable values from a numerical contradiction.

Multiply the first equation by 2: 4x + 2y = 14. Subtract from 4x + 2y = 16: (4x+2y) - (4x+2y) = 16 - 14, giving 0 = 2. This is a false statement. No ordered pair can satisfy both equations simultaneously, so the system is inconsistent with no solution. Options A and B claim specific solutions but neither satisfies both equations. Option C is incorrect because an identity not a contradiction would indicate infinite solutions.

Option B gives 5=0, a false numerical statement with no variables, confirming the system is inconsistent with no solution. Option D gives 3=-2, also a false statement confirming no solution. Option A gives 0=0, a true identity indicating the equations are dependent with infinitely many solutions. Option C gives 12=12, also a true identity with the same implication. Only false statements after full variable elimination confirm no solution.

Multiply the first by 2: 14x + 4y = 16. Add to -14x - 4y = -15: (14x - 14x) + (4y - 4y) = 16 + (-15), giving 0 = 1. This is a false statement confirming no solution. The second equation is nearly -2 times the first but with an inconsistent constant, making the lines parallel with no intersection point.

Add the equations: (5x - 5x) + (4y + 6y) = 13 + 1, giving 10y = 14, so y = 14/10 = 7/5. Option A gives 6/5, which would require 10y=12. Option C gives 8/5, requiring 10y=16. Option D gives 9/5, requiring 10y=18. Only 7/5 correctly results from adding the two equations.

The answer is True. Lines with equal slopes are parallel and never intersect. When you apply elimination to their equations, the variable terms cancel completely and you are left with a false numerical statement such as 0 = c where c is nonzero. This contradiction confirms no intersection exists and the system has no solution.

Add the equations to eliminate y: (2x+3y) + (4x-3y) = 11 + 1, giving 6x = 12, so x = 2. Substitute into 2x+3y=11: 4+3y=11, so 3y=7, y=7/3. The solution is (2, 7/3). Option A gives y=1 but substituting: 2(2)+3(1)=7≠11. Option C gives (1,3): 2+9=11 ✓ but 4-9=-5≠1. Option D gives (3,2): 6+6=12≠11.

Option A: rewrite 4x-2y=5 as y=2x-2.5, same slope as y=2x+1 but different intercept, giving contradiction. Option C: double first gives 2x-6y=4, subtract from 2x-6y=3 gives 0=-1, contradiction. Option D: double first gives 6x+2y=14, subtract from 6x+2y=15 gives 0=1, contradiction. Option B doubles to 4x+6y=22, identical to second equation — an identity with infinite solutions, not a contradiction.

Multiply the first equation by 2: 6x - 4y = 16. Subtract from 6x - 4y = 15: 0 = 15 - 16 = -1, a contradiction. No ordered pair satisfies both equations. The second equation is almost a multiple of the first but with an inconsistent constant, confirming the lines are parallel with no intersection. Options A and D incorrectly assume a solution exists. Option C would require the scaled equations to be identical.

The statement 0 = 5 is a numerical contradiction that is false for all variable values. When elimination removes all variables and produces a false statement, it proves the equations represent parallel lines with no intersection point. The system has no solution. Option A confuses contradiction with identity — 0=0 would indicate infinite solutions. Options B and D incorrectly extract variable values from a numerical statement.

The answer is True. The statement 0 = -3 is a false numerical equation — no variable values can make 0 equal -3. When elimination removes all variables and leaves a false constant statement, the two equations represent parallel lines that never intersect. The system is therefore inconsistent and has no solution.