Substitution Contradiction Quiz: Substitution Method Contradiction

The answer is False. The statement 10=10 is not a contradiction — it is a true identity. When substitution yields a true statement with no variables remaining, it means the two equations are multiples of each other and represent the same line. This produces infinitely many solutions, not a unique one. A unique solution requires finding a specific numerical value for each variable.

When substitution eliminates all variables and produces a false numerical statement such as 5=-2, it proves no ordered pair can satisfy both equations simultaneously. Option A finding x=y=0 would be a valid specific solution. Option B getting 0=0 is an identity indicating infinitely many solutions, not no solution. Option D obtaining specific x and y values indicates a unique solution. Only a false numerical statement after full variable elimination proves inconsistency.

The answer is True. Lines with the same slope are parallel. Parallel lines with different y-intercepts never intersect at any point. When you solve such a system algebraically, the variable terms always cancel and you are left with a false statement such as 3=7, confirming no solution exists. If the intercepts were also equal the lines would coincide, giving infinitely many solutions instead.

The statement -6=-5 is a false numerical equation — no variable values can make -6 equal to -5. When substitution eliminates all variables and produces a false statement, the system is inconsistent and has no solution. The lines are parallel and never intersect. Option A confuses false statements with identities. Options B and C incorrectly extract variable values from a numerical contradiction.

Multiply the first equation by 2: 2x-4y=8. The second equation is 2x-4y=10. The left sides are identical but the right sides differ, so subtracting gives 0=2, a contradiction. No ordered pair can satisfy both equations simultaneously. The lines are parallel with the same slope but different intercepts, confirming no solution.

Option A: 4x-2y=7 becomes y=2x-7/2, slope 2. First line has slope 2 and intercept 1. Same slope different intercepts — parallel, giving contradiction -7/2=1 after substitution. Option C: x-2y=9 becomes y=x/2-9/2, slope 1/2. First line has slope 1/2 and intercept -4. Different intercepts — parallel, giving 8=9 after substitution. Option D: 6x-2y=5 becomes y=3x-5/2, slope 3. First has slope 3 and intercept -2. Different intercepts — parallel, giving contradiction. Option B yields 6=6, an identity meaning dependent lines with infinite solutions.

The answer is True. Obtaining specific numerical values for both variables without any contradiction means exactly one ordered pair satisfies both equations. The pair (2,5) can be verified by substituting back into both original equations. This is the definition of a unique solution — one and only one point where the two lines intersect.

Substitute y into 9x+3y=12: 9x+3(-3x+5)=12, giving 9x-9x+15=12, so 15=12. This is a false statement and a contradiction. No ordered pair satisfies both equations simultaneously and the system has no solution. Options B, C, and D all claim specific solutions but substituting any of them into the original equations confirms none satisfy both simultaneously.

Compute 2y = 2(4-2x) = 8-4x. The second equation gives 9-4x. Setting equal: 8-4x = 9-4x, so -4x cancels from both sides leaving 8=9. This is a false statement confirming no solution. Option A gives 8=8, which would be an identity implying infinite solutions. Option B gives 9=9, also an identity. Option C gives 4=9, which is a contradiction but does not match the actual calculation.

Substitute y into 6x-2y=4: 6x-2(3x+1)=4, giving 6x-6x-2=4, so -2=4, a contradiction. Option B: substitute giving 4x+2(-2x+5)=10, so 10=10, an identity with infinite solutions. Option C: substitute giving 3x-(x-1)=5, so 2x+1=5, x=2, a unique solution. Option D: substitute giving x+(-x+4)=6, so 4=6, actually also a contradiction — but per the source the intended answer is A.

From x + y = 5, get y = 5 - x. Substitute into 2x + 2y = 11: 2x + 2(5-x) = 11, giving 2x + 10 - 2x = 11, so 10 = 11. This is a false statement — no values of x and y can make 10 equal 11. The system is inconsistent and has no solution. Options A, B, and C all claim specific solutions exist, but none can satisfy both equations simultaneously.

Option B gives 5=0, a false numerical statement with no variables, meaning the system is inconsistent and has no solution. Option D gives 2=-3, also a false statement confirming no solution. Option A gives 0=0, a true identity indicating the equations are dependent and have infinitely many solutions. Option C gives 7=7, also a true identity with the same implication. Only false statements confirm no solution.

Substitute y into 2x-y=-1: 2x-(x+2)=-1, giving x-2=-1, so x=1. Then y=1+2=3. Check: y=1+2=3 ✓ and 2(1)-3=-1 ✓. The unique solution is (1,3). Option A gives (2,4): check 2(2)-4=0≠-1. Option C incorrectly concludes no solution when one exists. Option D gives (0,2): check 2(0)-2=-2≠-1.

From 3x-y=2, get y=3x-2. Substitute into 6x-2y=5: 6x-2(3x-2)=5, giving 6x-6x+4=5, so 4=5. This is a contradiction. The second equation is a multiple of the first in structure but has an inconsistent constant, confirming the lines are parallel and the system has no solution.

Substitute y into 4x-2y=9: 4x-2(2x-4)=9, giving 4x-4x+8=9, so 8=9. This is a false statement and a contradiction. No ordered pair can satisfy both equations simultaneously, so the system has no solution. The lines are parallel with equal slopes but different intercepts. Options A, C, and D all claim specific solutions but none satisfy both equations.

The answer is True. Lines with equal slopes are parallel and never intersect. When you substitute one equation into the other, the variable terms cancel and you are left with a false numerical statement such as 3 = 7. This contradiction confirms the lines never meet and the system has no solution. If the intercepts were also equal the lines would be identical, giving infinitely many solutions instead.

Option C: substitute into 2x+2y=5 giving 2x+2(-x+1)=5, so 2=5, a contradiction. Option D: substitute into 6x-3y=2 giving 6x-3(2x-1)=2, so 3=2, a contradiction. Option A: substitute giving 2x-2(x-3)=6, so 6=6, an identity meaning infinite solutions not no solution. Option B yields a unique solution with specific x and y values.

Substitute y into 6x + 2y = 1: 6x + 2(3x+2) = 1, giving 6x + 6x + 4 = 1, so 12x = -3, x = -1/4. Then y = 3(-1/4) + 2 = -3/4 + 2 = 5/4. Check in both equations: y = 3(-1/4)+2 = 5/4 ✓ and 6(-1/4)+2(5/4) = -3/2+5/2 = 1 ✓. The unique solution is (-1/4, 5/4).

The statement 0 = 5 is a contradiction because 0 can never equal 5 regardless of any variable values. When substitution eliminates all variables and leaves a false numerical statement, it proves the two equations represent parallel lines that never intersect. The system is inconsistent and has no solution. Option A confuses contradiction with identity. Options B and D incorrectly interpret the result as yielding specific values.

The answer is True. The statement 3 = -1 is a contradiction — it is false for all values of x and y. When substitution eliminates the variables and produces a false numerical equation, it proves that no ordered pair can satisfy both equations at the same time. The system is therefore inconsistent with no solution.