Elimination Method Quiz: Elimination Impossible Statement

1) Use elimination on 2x + y = 7 and 4x + 2y = 16. What is the conclusion?

(x,y) = (2,3)

(x,y) = (3,1)

Infinitely many solutions

No solution

Multiply the first equation by 2 to align y: 4x + 2y = 14. Subtract this from 4x + 2y = 16 to eliminate both variables: (4x+2y) − (4x+2y) = 16 − 14 ⇒ 0 = 2, an impossibility. Hence the system is inconsistent with no solution.

Explanation

Multiply the first equation by 2 to align y: 4x + 2y = 14. Subtract this from 4x + 2y = 16 to eliminate both variables: (4x+2y) − (4x+2y) = 16 − 14 ⇒ 0 = 2, an impossibility. Hence the system is inconsistent with no solution.

2) If elimination produces 0 = −3, then the system has no solution.

True

False

A false statement (like 0 = −3) after eliminating variables means the two equations are inconsistent, so there is no ordered pair satisfying both.

Explanation

A false statement (like 0 = −3) after eliminating variables means the two equations are inconsistent, so there is no ordered pair satisfying both.

3) After scaling and subtracting, a system reduces to 0 = 5. The correct conclusion is ____.

A numerical contradiction indicates parallel, non-intersecting lines. Therefore, the system has no solution.

Explanation

A numerical contradiction indicates parallel, non-intersecting lines. Therefore, the system has no solution.

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4) Apply elimination to 3x − 2y = 8 and 6x − 4y = 15. What is the conclusion?

Unique solution

No solution

Infinitely many solutions

(x,y) = (2,−1)

Multiply the first equation by 2: 6x − 4y = 16. Subtract from the second: (6x − 4y) − (6x − 4y) = 15 − 16 ⇒ 0 = −1, a contradiction. Hence no solution.

Explanation

Multiply the first equation by 2: 6x − 4y = 16. Subtract from the second: (6x − 4y) − (6x − 4y) = 15 − 16 ⇒ 0 = −1, a contradiction. Hence no solution.

5) Which systems are inconsistent (will yield an impossibility under elimination)? Select all that apply.

Y = 2x + 1 and 4x − 2y = 5

2x + 3y = 11 and 4x + 6y = 22

X − 3y = 2 and 2x − 6y = 3

3x + y = 7 and 6x + 2y = 15

X + 2y = 4 and 2x + 4y = 9

A: Put 4x − 2y = 5 into slope form: −2y = −4x + 5 ⇒ y = 2x − 2.5, parallel to y = 2x + 1 (same slope, different intercept) ⇒ contradiction. C: Double the first gives 2x − 6y = 4; subtract from 2x − 6y = 3 ⇒ 0 = −1 ⇒ contradiction. D: Double the first gives 6x + 2y = 14; subtract from 6x + 2y = 15 ⇒ 0 = 1 ⇒ contradiction. E: Double the first gives 2x + 4y = 8; subtract from 2x + 4y = 9 ⇒ 0 = 1 ⇒ contradiction. B are multiples (identity), not a contradiction.

Explanation

A: Put 4x − 2y = 5 into slope form: −2y = −4x + 5 ⇒ y = 2x − 2.5, parallel to y = 2x + 1 (same slope, different intercept) ⇒ contradiction. C: Double the first gives 2x − 6y = 4; subtract from 2x − 6y = 3 ⇒ 0 = −1 ⇒ contradiction. D: Double the first gives 6x + 2y = 14; subtract from 6x + 2y = 15 ⇒ 0 = 1 ⇒ contradiction. E: Double the first gives 2x + 4y = 8; subtract from 2x + 4y = 9 ⇒ 0 = 1 ⇒ contradiction. B are multiples (identity), not a contradiction.

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6) Solve using elimination: 2x + 3y = 11 and 4x − 3y = 1. What is (x,y)?

(2,1)

(2,7/3)

(1,3)

(3,2)

Add the equations to eliminate y: (2x+3y) + (4x−3y) = 11 + 1 ⇒ 6x = 12 ⇒ x = 2. Substitute into 2x + 3y = 11 ⇒ 4 + 3y = 11 ⇒ 3y = 7 ⇒ y = 7/3. So (x,y) = (2, 7/3).

Explanation

Add the equations to eliminate y: (2x+3y) + (4x−3y) = 11 + 1 ⇒ 6x = 12 ⇒ x = 2. Substitute into 2x + 3y = 11 ⇒ 4 + 3y = 11 ⇒ 3y = 7 ⇒ y = 7/3. So (x,y) = (2, 7/3).

7) If two lines have equal slopes and different intercepts, elimination will yield a contradiction.

True

False

Equal slopes with different intercepts are parallel lines. Eliminating variables yields a false statement like 0 = c (c ≠ 0), confirming no solution.

Explanation

Equal slopes with different intercepts are parallel lines. Eliminating variables yields a false statement like 0 = c (c ≠ 0), confirming no solution.

8) Eliminate x in 5x + 4y = 13 and −5x + 6y = 1 by adding. The resulting y equals ____.

Add equations: (5x−5x) + (4y+6y) = 13 + 1 ⇒ 10y = 14 ⇒ y = 14/10 = 7/5.

Explanation

Add equations: (5x−5x) + (4y+6y) = 13 + 1 ⇒ 10y = 14 ⇒ y = 14/10 = 7/5.

Submit

9) Check consistency: 7x + 2y = 8 and −14x − 4y = −15. What is the conclusion?

No solution

(x,y) = (1,1)

Unique solution

Infinitely many solutions

Multiply the first by 2: 14x + 4y = 16. Add to the second: (14x+4y) + (−14x−4y) = 16 + (−15) ⇒ 0 = 1, impossible. Hence no solution.

Explanation

Multiply the first by 2: 14x + 4y = 16. Add to the second: (14x+4y) + (−14x−4y) = 16 + (−15) ⇒ 0 = 1, impossible. Hence no solution.

10) Elimination reduces a system to one of the following. Which outcomes imply no solution? Select all that apply.

0 = 0

5 = 0

12 = 12

3 = −2

Y = 4

Contradictions like 5 = 0 or 3 = −2 mean inconsistency. Identities (0 = 0, 12 = 12) indicate dependent equations. A statement like y = 4 leads to a unique solution after back‑substitution.

Explanation

Contradictions like 5 = 0 or 3 = −2 mean inconsistency. Identities (0 = 0, 12 = 12) indicate dependent equations. A statement like y = 4 leads to a unique solution after back‑substitution.

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11) Solve using elimination: 3x + 2y = 14 and 5x − 2y = 6. What is (x,y)?

(2,4)

(3,2)

(5/2, 13/4)

(5,−1)

Add to eliminate y: (3x+2y) + (5x−2y) = 14 + 6 ⇒ 8x = 20 ⇒ x = 20/8 = 5/2. Substitute into 3x + 2y = 14 ⇒ 3(5/2) + 2y = 14 ⇒ 15/2 + 2y = 14 ⇒ 2y = 13/2 ⇒ y = 13/4. Thus (x,y) = (5/2, 13/4).

Explanation

Add to eliminate y: (3x+2y) + (5x−2y) = 14 + 6 ⇒ 8x = 20 ⇒ x = 20/8 = 5/2. Substitute into 3x + 2y = 14 ⇒ 3(5/2) + 2y = 14 ⇒ 15/2 + 2y = 14 ⇒ 2y = 13/2 ⇒ y = 13/4. Thus (x,y) = (5/2, 13/4).

12) After multiplying and subtracting, the system becomes 0 = −12. The proper conclusion is ____.

Since the eliminated equation yields a false statement, the system is inconsistent and has no solution.

Explanation

Since the eliminated equation yields a false statement, the system is inconsistent and has no solution.

Submit

13) Which pairs will lead to an impossibility under elimination? Select all that apply.

X + y = 4 and 2x + 2y = 8

X − 2y = 1 and 2x − 4y = 5

3x + 6y = 7 and x + 2y = 3

2x + y = 5 and 4x + 2y = 10

4x − y = 8 and 8x − 2y = 17

B: Double first gives 2x − 4y = 2 vs 5 ⇒ subtract: 0 = 3 contradiction. C: Multiply the second by 3: 3x + 6y = 9 vs 7 ⇒ subtract: 0 = −2 contradiction. E: Double first gives 8x − 2y = 16 vs 17 ⇒ subtract: 0 = 1 contradiction. A and D are multiples (identities).

Explanation

B: Double first gives 2x − 4y = 2 vs 5 ⇒ subtract: 0 = 3 contradiction. C: Multiply the second by 3: 3x + 6y = 9 vs 7 ⇒ subtract: 0 = −2 contradiction. E: Double first gives 8x − 2y = 16 vs 17 ⇒ subtract: 0 = 1 contradiction. A and D are multiples (identities).

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14) If coefficients of x and y in two equations are proportional but constants are not, elimination yields an impossibility.

True

False

Proportional coefficients with mismatched constants represent parallel lines. Eliminating variables wipes them out and leaves a false constant statement, confirming no solution.

Explanation

Proportional coefficients with mismatched constants represent parallel lines. Eliminating variables wipes them out and leaves a false constant statement, confirming no solution.

15) Analyze with elimination: 3x + 5y = 1 and 6x + 10y = 3. What is the conclusion?

No solution

Unique solution

(x,y) = (1,−2)

(x,y) = (0,1)

Multiply the first equation by 2: 6x + 10y = 2. Subtract from 6x + 10y = 3: (6x+10y) − (6x+10y) = 3 − 2 ⇒ 0 = 1, impossible. Hence no solution.

Explanation

Multiply the first equation by 2: 6x + 10y = 2. Subtract from 6x + 10y = 3: (6x+10y) − (6x+10y) = 3 − 2 ⇒ 0 = 1, impossible. Hence no solution.

16) Solve using elimination: 4x − 3y = 13 and 2x + 3y = 1. What is (x,y)?

(5,−1)

(7/3, −11/9)

(3,−2)

(2,−1)

Add equations to eliminate y: (4x−3y) + (2x+3y) = 13 + 1 ⇒ 6x = 14 ⇒ x = 7/3. Substitute into 2x + 3y = 1 ⇒ 2(7/3) + 3y = 1 ⇒ 14/3 + 3y = 1 ⇒ 3y = −11/3 ⇒ y = −11/9. Thus (x,y) = (7/3, −11/9).

Explanation

Add equations to eliminate y: (4x−3y) + (2x+3y) = 13 + 1 ⇒ 6x = 14 ⇒ x = 7/3. Substitute into 2x + 3y = 1 ⇒ 2(7/3) + 3y = 1 ⇒ 14/3 + 3y = 1 ⇒ 3y = −11/3 ⇒ y = −11/9. Thus (x,y) = (7/3, −11/9).

17) Eliminate y in 2x + y = 9 and 5x − y = 6 by adding. The resulting x equals ____.

Add equations: (2x+ y) + (5x − y) = 9 + 6 ⇒ 7x = 15 ⇒ x = 15/7.

Explanation

Add equations: (2x+ y) + (5x − y) = 9 + 6 ⇒ 7x = 15 ⇒ x = 15/7.

Submit

18) When elimination yields a contradiction like 0 = 4, which statements must be true? Select all that apply.

The lines are parallel.

The slopes are equal.

The y‑intercepts are different.

There is no solution.

There is exactly one solution.

A contradiction means the two lines are parallel: same slope, different intercepts, so they never meet and the system has no solution.

Explanation

A contradiction means the two lines are parallel: same slope, different intercepts, so they never meet and the system has no solution.

Submit

19) If elimination gives 12 = 12 after scaling, the system has a unique solution.

True

False

A true identity indicates the equations are dependent (same line), which corresponds to infinitely many solutions, not a unique solution.

Explanation

A true identity indicates the equations are dependent (same line), which corresponds to infinitely many solutions, not a unique solution.

20) Check consistency: x − 2y = 4 and 2x − 4y = 9. What is the conclusion?

Unique solution

No solution

(x,y) = (2,−1)

Infinitely many solutions

Double the first: 2x − 4y = 8. Compare with 2x − 4y = 9. Subtract to eliminate variables: 0 = 1, a contradiction, so there is no solution.

Explanation

Double the first: 2x − 4y = 8. Compare with 2x − 4y = 9. Subtract to eliminate variables: 0 = 1, a contradiction, so there is no solution.