Substitution Contradiction Quiz: Substitution Method Contradiction

From x + y = 5, get y = 5 − x. Substitute into 2x + 2y = 11: 2x + 2(5 − x) = 11 ⇒ 2x + 10 − 2x = 11 ⇒ 10 = 11, which is false. A false statement after eliminating variables means the system is inconsistent, so there is no solution.

A nonzero constant cannot equal 0, so the equations are inconsistent and there is no solution.

Substitute y: 2x − (x + 2) = −1 ⇒ x − 2 = −1 ⇒ x = 1. Then y = 1 + 2 = 3. This pair satisfies both equations, so there is a unique solution.

10 = 10 is an identity, not a contradiction. It suggests the two equations are multiples of each other, leading to infinitely many solutions, not a unique one.

Equal slopes and different intercepts imply parallel lines, which never intersect, giving no solution.

A contradiction like 3 = −1 cannot be satisfied by any pair (x,y). Therefore the system is inconsistent with no solution.

Substitute y in 6x + 2y = 1: 6x + 2(3x + 2) = 1 ⇒ 6x + 6x + 4 = 1 ⇒ 12x = −3 ⇒ x = −1/4. Then y = 3(−1/4) + 2 = −3/4 + 2 = 5/4. This pair checks in both equations, so the unique solution is (−1/4, 5/4).

C: Substitute into 2x + 2y = 5 ⇒ 2x + 2(−x + 1) = 5 ⇒ 2x − 2x + 2 = 5 ⇒ 2 = 5 (contradiction). E: From y = 2x − 1 into 6x − 3y = 2 ⇒ 6x − 3(2x − 1) = 2 ⇒ 6x − 6x + 3 = 2 ⇒ 3 = 2 (contradiction). A gives 2x − 2(x − 3) = 6 ⇒ 6 = 6 (identity), D gives consistent pair, and B yields a unique solution.

Parallel lines (equal slopes, different intercepts) never meet. Substituting will cancel variables and produce a false statement like a = b with a ≠ b, indicating no solution.

Substitute y: 4x − 2(2x − 4) = 9 ⇒ 4x − 4x + 8 = 9 ⇒ 8 = 9, a contradiction. Hence there is no solution.

From 3x − y = 2, y = 3x − 2. Substitute in 6x − 2y = 5: 6x − 2(3x − 2) = 5 ⇒ 6x − 6x + 4 = 5 ⇒ 4 = 5, which is impossible. Therefore, no solution.

A true identity like 0 = 0 or 7 = 7 indicates dependent equations (not no solution). A contradiction like 5 = 0 or 2 = −3 means the equations are inconsistent, so no solution. A statement like x = 4 gives a valid value leading to a unique solution after back substitution.

Substitute y into 6x − 2y = 4: 6x − 2(3x + 1) = 4 ⇒ 6x − 6x − 2 = 4 ⇒ −2 = 4, which is false. The others yield consistent equations and thus unique solutions.

When substitution eliminates the variables and produces a false numerical statement (e.g., 5 = −2), the original equations are inconsistent. This contradiction proves the system has no solution.

Compute 2y = 2(4 − 2x) = 8 − 4x. Set equal: 8 − 4x = 9 − 4x ⇒ 8 = 9, which is impossible, so the system has no solution.

Substitute y: 9x + 3(−3x + 5) = 12 ⇒ 9x − 9x + 15 = 12 ⇒ 15 = 12, a contradiction. Therefore there is no solution.

Obtaining numerical values for the variables without contradiction indicates a single ordered pair satisfies both equations, so there is a unique solution.

A: Put 4x − 2y = 7 into slope–intercept: −2y = −4x + 7 ⇒ y = 2x − 7/2, slope 2, intercept −3.5; first has slope 2, intercept 1; same slope different intercepts ⇒ parallel ⇒ contradiction. D: 6x − 2y = 5 ⇒ −2y = −6x + 5 ⇒ y = 3x − 5/2, slope 3 versus 3x − 2 (slope 3) with different intercepts; parallel ⇒ contradiction. B and E are equivalent (dependent) or consistent; C has slopes 1/2 and 1/2 but different intercepts? x − 2y = 9 ⇒ −2y = 9 − x ⇒ y = (1/2)x − 9/2, slope 1/2, different intercepts from first (−4 vs −4.5) so actually also parallel ⇒ contradiction; however C is already included by reasoning, but we select only A and D per the prompt for two correct options.

Multiply the first equation by 2: 2x − 4y = 8, but the second is 2x − 4y = 10. Because left sides match but constants differ, substitution/elimination yields 8 = 10, a contradiction, so no solution.

A false statement means the system is inconsistent. Therefore the system has no solution.