Solving Systems of Linear Equations - Algebra 1

1. Identify the solution: Solve the system $y=\frac{1}{2}x+2$ and $y=-\frac{1}{2}x+4$.

   $(2,3)$. Set equations equal: $\frac{1}{2}x+2=-\frac{1}{2}x+4$ gives $x=2$.

2. Identify the solution: Solve the system $y=2x$ and $y=2x+6$.

   No solution. Same slope but different y-intercepts means parallel lines.

3. Identify the solution: Solve the system $y=2x+1$ and $y=11$.

   $(5,11)$. Substitute $y=11$ into $y=2x+1$ to get $x=5$.

4. Identify the solution: Solve the system $x+y=1$ and $x=4$.

   $(4,-3)$. Substitute $x=4$ into first equation to get $y=-3$.

5. Identify the solution: Solve the system $x+4y=16$ and $x+4y=0$.

   No solution. Same left sides but different right sides creates a contradiction.

6. Identify the solution: Solve the system $2x-y=7$ and $x+y=5$.

   $(4,1)$. Add equations to eliminate $y$: $3x=12$, so $x=4$.

7. Identify the solution type: Solve the system $x+y=7$ and $x+y=9$.

   No solution. Same left sides but different right sides creates a contradiction.

8. Identify the solution: Solve the system $2x+5y=20$ and $x=5$.

   $(5,2)$. Substitute $x=5$ into first equation to get $y=2$.

9. What is the standard form of a linear equation used often for elimination?

   $Ax+By=C$ with $A,B,C$ real and $A$ and $B$ not both $0$. Standard form is useful for elimination method calculations.

10. Identify the solution: Solve the system $x- y=1$ and $x+y=7$.

    $(4,3)$. Add equations to eliminate $y$: $2x=8$, so $x=4$.

11. Identify the solution: Solve the system $y=3x-2$ and $y=3x+5$.

    No solution. Same slope but different y-intercepts means parallel lines.

12. Identify the solution: Solve the system $3x+y=12$ and $y=3$.

    $(3,3)$. Substitute $y=3$ into first equation to get $x=3$.

13. Identify the solution: Solve the system $y=x+2$ and $y=-2x+8$.

    $(2,4)$. Set equations equal: $x+2=-2x+8$ gives $x=2$.

14. What slope condition guarantees exactly one solution for $y=m_1x+b_1$ and $y=m_2x+b_2$?

    $m_1\ne m_2$. Different slopes guarantee the lines will intersect once.

15. What method solves a system by solving one equation for a variable and substituting?

    The substitution method. Replace one variable with an expression from the other equation.

16. Identify the solution type: Solve the system $y=\frac{2}{3}x-1$ and $3y=2x-3$.

    Infinitely many solutions. The second equation simplifies to the same as the first.

17. What slope condition guarantees no solution for $y=m_1x+b_1$ and $y=m_2x+b_2$?

    $m_1=m_2$ and $b_1\ne b_2$. Same slope, different intercepts means parallel lines.

18. What does it mean if elimination produces a false statement like $0=5$?

    No solution (inconsistent system). A contradiction indicates parallel lines that never meet.

19. Identify the solution: Solve the system $x+y=9$ and $y=9-x$.

    Infinitely many solutions. The second equation rearranges to match the first exactly.

20. Identify the solution: Solve the system $2x+y=0$ and $x-y=3$.

    $(1,-2)$. Add equations to eliminate $y$: $3x=3$, so $x=1$.

21. What is the key goal in elimination when combining two equations?

    Make one variable cancel to $0$ when adding equations. Create opposite coefficients so one variable eliminates completely.

22. What is an independent system of linear equations?

    A system with exactly one solution (intersecting lines). The lines intersect at exactly one point.

23. Identify the solution: Solve the system $x+y=10$ and $x-y=2$.

    $(6,4)$. Add equations to eliminate $y$: $2x=12$, so $x=6$.

24. What is the first step of substitution when one equation is not solved for a variable?

    Solve one equation for $x$ or for $y$. Isolate one variable to substitute into the other equation.

25. What method solves a system by graphing and finding the intersection point?

    The graphing method. Plot both lines and find where they cross.

26. Identify the solution: Solve the system $y=5$ and $y=-2x+1$.

    $(-2,5)$. Substitute $y=5$ into second equation to get $x=-2$.

27. What is the solution to a system of two linear equations in two variables?

    The ordered pair $(x, y)$ that makes both equations true. The point that satisfies both linear equations simultaneously.

28. Identify the solution: Solve the system $x= -3$ and $y=5$.

    $(-3,5)$. Both variables are directly given as constants.

29. What does it mean if a system of two linear equations has exactly one solution?

    The lines intersect at exactly one point $(x, y)$. This describes the geometric interpretation of intersecting lines.

30. Identify the solution: Solve the system $2x-y=4$ and $y=2x-4$.

    Infinitely many solutions. Both equations represent the same line when rearranged.

31. What slope condition guarantees infinitely many solutions for $y=m_1x+b_1$ and $y=m_2x+b_2$?

    $m_1=m_2$ and $b_1=b_2$. Same slope and intercept means identical lines.

32. Identify the solution: Solve the system $x+y=0$ and $y=2x$.

    $(0,0)$. Substitute $y=2x$ into first equation: $x+2x=0$.

33. Identify the solution: Solve the system $x+3y=3$ and $x=3$.

    $(3,0)$. Substitute $x=3$ into first equation to get $y=0$.

34. What is a consistent system of linear equations?

    A system with at least one solution. Either one unique solution or infinitely many solutions exist.

35. What does it mean if elimination produces a true statement like $0=0$?

    Infinitely many solutions (dependent system). The equations are equivalent, representing the same line.

36. Identify the solution: Solve the system $x+y=7$ and $2x+2y=14$.

    Infinitely many solutions. The second equation is twice the first, so they're equivalent.

37. Identify the solution: Solve the system $x+y=6$ and $y=2$.

    $(4,2)$. Substitute $y=2$ into first equation to get $x=4$.

38. Identify the solution: Solve the system $x-2y=0$ and $x+y=9$.

    $(6,3)$. Subtract equations to eliminate $x$: $-3y=-9$, so $y=3$.

39. Identify the solution: Solve the system $y=-2x+1$ and $2y=-4x+2$.

    Infinitely many solutions. The second equation is equivalent to the first when simplified.

40. What is the approximate solution from a graph of a system of lines?

    The estimated intersection point $(x, y)$ read from the graph. Reading coordinates from graph gives approximate values.

41. What does it mean if a system of two linear equations has no solution?

    The lines are parallel and never intersect. Parallel lines have the same slope but different y-intercepts.

42. What is an inconsistent system of linear equations?

    A system with no solution. The lines are parallel and never meet.

43. Identify the solution: Solve the system $x+y=9$ and $y=7-x$.

    No solution. Different y-intercepts with same slope means parallel lines.

44. Identify the solution: Solve the system $2x+3y=12$ and $2x+3y=12$.

    Infinitely many solutions. Both equations are identical, so all points satisfy both.

45. Identify the solution: Solve the system $y=4x$ and $y=4x-8$.

    No solution. Same slope but different y-intercepts means parallel lines.

46. Identify the solution: Solve the system $2x+3y=13$ and $y=3$.

    $(2,3)$. Substitute $y=3$ into first equation to get $x=2$.

47. Identify the solution: Solve the system $4x+y=1$ and $y=1-4x$.

    Infinitely many solutions. Both equations represent the same line when rearranged.

48. What method solves a system by adding scaled equations to eliminate a variable?

    The elimination (linear combination) method. Multiply equations to make coefficients opposite, then add.

49. Identify the solution: Solve the system $x-3y=9$ and $x=0$.

    $(0,-3)$. Substitute $x=0$ into first equation to get $y=-3$.

50. Identify the solution: Solve the system $y=1-x$ and $y=3-x$.

    No solution. Same slope but different y-intercepts means parallel lines.

51. Identify the solution: Solve the system $3x-3y=6$ and $x-y=2$.

    Infinitely many solutions. The second equation is equivalent to the first when simplified.

52. Identify the solution: Solve the system $x+y=5$ and $x+2y=8$.

    $(2,3)$. Subtract equations to eliminate $x$: $-y=-3$, so $y=3$.

53. What is a dependent system of linear equations?

    A system with infinitely many solutions (same line). The equations represent the same line.

54. What does it mean if a system of two linear equations has infinitely many solutions?

    The equations represent the same line. Both equations describe identical lines with all points in common.

55. Identify the solution type: Solve the system $2x+3y=12$ and $4x+6y=10$.

    No solution. The second equation simplifies to $2x+3y=5$, contradicting the first.

56. Identify the solution: Solve the system $x+2y=12$ and $x+2y=10$.

    No solution. Same left sides but different right sides creates a contradiction.

57. Identify the solution: Solve the system $x+2y=8$ and $x=2$.

    $(2,3)$. Substitute $x=2$ into first equation to get $y=3$.

58. Identify the solution: Solve the system $y=-x+4$ and $y=x$.

    $(2,2)$. Set equations equal: $-x+4=x$ gives $x=2$.

59. What is the slope-intercept form that is convenient for graphing a line?

    $y=mx+b$. Shows slope $m$ and y-intercept $b$ directly.

60. Identify the solution: Solve the system $2x+y=9$ and $x-y=3$.

    $(4,1)$. Add equations to eliminate $y$: $3x=12$, so $x=4$.

61. Identify the solution: Solve the system $x=1$ and $2x+y=10$.

    $(1,8)$. Substitute $x=1$ into second equation to get $y=8$.

Study these flashcards