Systems of linear equations Test

{"questions":[{"content":"For $2x+3y=19$ and $x-y=2$, enter x. [[input-linear-systems-test-1]]","widgets":{"input-linear-systems-test-1":{"type":"input","answer":"5"}},"hints":[],"explanation":"x = y + 2 gives 5y + 4 = 19; y = 3 and x = 5.","id":"linear-systems-test-1","coverage_topics":["aa-hl-systems-of-linear-equations"],"correct_feedback":"x = y + 2 gives 5y + 4 = 19; y = 3 and x = 5.","incorrect_feedback":"x = y + 2 gives 5y + 4 = 19; y = 3 and x = 5.","evidence_kind":"objective-concept-test"},{"content":"A museum sold 20 tickets. Adult tickets cost 8 units and child tickets cost 5 units; the total was 136 units. Enter the number of adult tickets. [[input-linear-systems-test-2]]","widgets":{"input-linear-systems-test-2":{"type":"input","answer":"12"}},"hints":[],"explanation":"Let a be the adult count and c the child count. Then a + c = 20 and 8a + 5c = 136; subtracting 5(a + c) gives 3a = 36.","id":"linear-systems-test-2","coverage_topics":["aa-hl-systems-of-linear-equations"],"correct_feedback":"Let a be the adult count and c the child count. Then a + c = 20 and 8a + 5c = 136; subtracting 5(a + c) gives 3a = 36.","incorrect_feedback":"Translate the total number and total cost into two equations for the same ticket counts.","evidence_kind":"objective-concept-test"},{"content":"Classify $3x-6y=9$ and $x-2y=4$.\u003Cbr\u003E[[choice-linear-systems-test-3]]","widgets":{"choice-linear-systems-test-3":{"type":"choice","options":["Infinitely many solutions","No solutions","One solution"],"answer":[1],"choice_feedback":["The constants are not in the same ratio.","Three times the second has right side 12, conflicting with 9.","Proportional coefficients with conflicting constants cannot intersect."]}},"hints":[],"explanation":"Three times the second has right side 12, conflicting with 9.","id":"linear-systems-test-3","coverage_topics":["aa-hl-systems-of-linear-equations"],"correct_feedback":"Three times the second has right side 12, conflicting with 9.","incorrect_feedback":"Three times the second has right side 12, conflicting with 9.","evidence_kind":"objective-concept-test"},{"content":"The equations are $x+y=3$ and $2x+ky=6$. Which classification is correct?\u003Cbr\u003E[[choice-linear-systems-test-4]]","widgets":{"choice-linear-systems-test-4":{"type":"choice","options":["Infinitely many solutions if $k=2$; one solution otherwise","No solution if $k=2$; one solution otherwise","Infinitely many solutions for every k"],"answer":[0],"choice_feedback":["Subtract twice the first equation: (k − 2)y = 0. If k = 2 the equations coincide; otherwise y = 0 and x = 3.","At k = 2 the constants scale with the coefficients, so there is no contradiction.","The second coefficient row is proportional to the first only when k = 2."]}},"hints":[],"explanation":"Subtract twice the first equation: (k − 2)y = 0. If k = 2 the equations coincide; otherwise y = 0 and x = 3.","id":"linear-systems-test-4","coverage_topics":["aa-hl-systems-of-linear-equations"],"correct_feedback":"Subtract twice the first equation: (k − 2)y = 0. If k = 2 the equations coincide; otherwise y = 0 and x = 3.","incorrect_feedback":"Subtract twice the first equation: (k − 2)y = 0. If k = 2 the equations coincide; otherwise y = 0 and x = 3.","evidence_kind":"objective-concept-test"},{"content":"Solve $x+y+z=10$, $x-y+z=4$, $2x+y-z=5$: enter x. [[input-linear-systems-test-5]]","widgets":{"input-linear-systems-test-5":{"type":"input","answer":"3"}},"hints":[],"explanation":"y = 3, x + z = 7, 2x − z = 2; hence 3x = 9.","id":"linear-systems-test-5","coverage_topics":["aa-hl-systems-of-linear-equations"],"correct_feedback":"y = 3, x + z = 7, 2x − z = 2; hence 3x = 9.","incorrect_feedback":"y = 3, x + z = 7, 2x − z = 2; hence 3x = 9.","evidence_kind":"objective-concept-test"},{"content":"A three-variable system reduces to $x+y=2$, $z=1$ and $0=0$. Which statement follows?\u003Cbr\u003E[[choice-linear-systems-test-6]]","widgets":{"choice-linear-systems-test-6":{"type":"choice","options":["There is no solution because one equation vanished","There is a one-parameter family of solutions","The unique solution is (2,0,1)"],"answer":[1],"choice_feedback":["The vanished equation is an identity, not a contradiction.","Choose y freely, set x = 2 − y and z = 1. The identity adds no restriction.","That is one solution, but (1,1,1) also satisfies every equation."]}},"hints":[],"explanation":"Choose y freely, set x = 2 − y and z = 1. The identity adds no restriction.","id":"linear-systems-test-6","coverage_topics":["aa-hl-systems-of-linear-equations"],"correct_feedback":"Choose y freely, set x = 2 − y and z = 1. The identity adds no restriction.","incorrect_feedback":"Choose y freely, set x = 2 − y and z = 1. The identity adds no restriction.","evidence_kind":"objective-concept-test"},{"content":"Which pair solves both $5x-y=8$ and $x+2y=6$?\u003Cbr\u003E[[choice-linear-systems-test-7]]","widgets":{"choice-linear-systems-test-7":{"type":"choice","options":["$(2,2)$","$(1,3)$","$(2,-2)$"],"answer":[0],"choice_feedback":["Substitution gives 10 − 2 = 8 and 2 + 4 = 6.","At (1,3), the two left sides are 2 and 7, so neither equation holds.","The first gives 12 and the second −2."]}},"hints":[],"explanation":"Substitution gives 10 − 2 = 8 and 2 + 4 = 6.","id":"linear-systems-test-7","coverage_topics":["aa-hl-systems-of-linear-equations"],"correct_feedback":"Substitution gives 10 − 2 = 8 and 2 + 4 = 6.","incorrect_feedback":"Substitution gives 10 − 2 = 8 and 2 + 4 = 6.","evidence_kind":"objective-concept-test"},{"content":"A three-variable system reduces to x+y=2, z=1 and x+y=5. Classify it.\u003Cbr\u003E[[choice-linear-systems-test-8]]","widgets":{"choice-linear-systems-test-8":{"type":"choice","options":["Infinitely many solutions","One solution","No solutions"],"answer":[2],"choice_feedback":["A free variable cannot repair inconsistent constraints.","The value z = 1 does not resolve the conflict in x + y.","The two constraints on x + y contradict one another."]}},"hints":[],"explanation":"The two constraints on x + y contradict one another.","id":"linear-systems-test-8","coverage_topics":["aa-hl-systems-of-linear-equations"],"correct_feedback":"The two constraints on x + y contradict one another.","incorrect_feedback":"The two constraints on x + y contradict one another.","evidence_kind":"objective-concept-test"}],"mix":0,"tinkumo_module_mode":"test"}