Solve Systems of Linear Equations - 8th Grade Math

1. What condition on the equations gives infinitely many solutions for a linear system?

   Equivalent equations (the same line), such as multiples of each other. Same line means every point on it satisfies both equations.

2. What condition on slopes and intercepts gives exactly one solution for two lines?

   Different slopes (not parallel), so the lines intersect once. Non-parallel lines must cross at exactly one point.

3. What is the solution to the system $x=3$ and $y=-2$ (by inspection)?

   $(3,-2)$. The values are given directly; verify they work in both.

4. Solve by elimination: $3x+2y=12$ and $3x-2y=4$.

   $( \frac{8}{3},2)$. Add equations: $6x=16$, so $x=\frac{8}{3}$; then $y=2$.

5. What is the slope-intercept form used for graphing a line?

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

6. What is the solution type by inspection: $2x-3y=6$ and $4x-6y=12$?

   Infinitely many solutions. Second equation is $2×$ the first, so same line.

7. Solve by elimination: $x+y=7$ and $x-y=1$.

   $(4,3)$. Add equations: $2x=8$, so $x=4$; subtract: $2y=6$, so $y=3$.

8. What is the solution type for $y=2x+3$ and $y=2x-1$?

   No solution (parallel lines). Same slope $m=2$ but different $y$-intercepts.

9. What is the solution to the system $2x+y=8$ and $y=2$ (by inspection)?

   $(3,2)$. Substitute $y=2$ into first: $2x+2=8$, so $x=3$.

10. Which method is best if $x$-coefficients are opposites, like $x+2y=5$ and $-x+y=1$?

    Elimination by adding. Adding equations directly eliminates $x$: $(0)x + 3y = 6$.

11. Solve by substitution: $y=2x$ and $x+y=9$.

    $(3,6)$. Substitute $y=2x$ into $x+y=9$: $x+2x=9$, so $3x=9$, $x=3$.

12. What is the solution to the system: $y=2x+1$ and $y=x+4$?

    $(3,7)$. Set equal: $2x+1=x+4$, so $x=3$ and $y=7$.

13. Solve by elimination: $2x+y=9$ and $-2x+y=1$.

    $(2,5)$. Add equations to eliminate $x$: $2y=10$, so $y=5$.

14. What does it mean if a system has exactly one solution?

    The lines intersect at exactly one point $(x,y)$. This is the unique solution where both equations are satisfied.

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

    The ordered pair $(x,y)$ that makes both equations true. It's the point where both lines intersect on a graph.

16. What does it mean if two lines are parallel on a graph (same slope, different intercepts)?

    The system has no solution. Parallel lines never intersect, so no point satisfies both.

17. Solve by elimination: $2x+3y=12$ and $4x+6y=30$.

    No solution. Second has twice the coefficients but different constant, parallel.

18. Identify the number of solutions: $y=3x-2$ and $y=3x+4$.

    No solution. Same slope (3) but different y-intercepts means parallel.

19. What is the solution type by inspection: $x-2y=1$ and $2x-4y=5$?

    No solution. Second has $2×$ left side but not $2×$ right side.

20. What is the elimination method for solving a system used for?

    Adding or subtracting equations to eliminate one variable. Combine equations to cancel out one variable first.

21. What is the first step to solve by elimination when no variable cancels?

    Multiply one or both equations to create opposite coefficients. Make coefficients opposites so they cancel when added.

22. Identify the solution by inspection: $y=-x+4$ and $2y=-2x+8$.

    Infinitely many solutions. Second equation is first multiplied by 2, so they're the same line.

23. What is the approximate solution if two graphed lines intersect at $(2.1,3.9)$?

    $(2.1,3.9)$. The intersection point is the graphical solution.

24. What does it mean if you get a false statement like $0=5$ after elimination?

    No solution. The equations contradict each other (parallel lines).

25. What is the approximate solution where $y=x+1$ meets $y=-x+5$?

    $(2,3)$. Set equal: $x+1=-x+5$; solve to get $x=2$, $y=3$.

26. Solve by elimination: $2x+3y=12$ and $4x+6y=24$.

    Infinitely many solutions. Second equation is first multiplied by 2, same line.

27. Which method is best if one equation is already solved for a variable, like $y=3x-2$?

    Substitution. Already isolated variables save a step in substitution.

28. What is the solution to the system: $y=-x+2$ and $y=x-4$?

    $(3,-1)$. Set equal: $-x+2=x-4$, so $2x=6$, thus $x=3$ and $y=-1$.

29. What is the solution to the system: $x+y=5$ and $x-y=1$?

    $(3,2)$. Add equations: $2x=6$, so $x=3$; subtract: $2y=4$, so $y=2$.

30. What are the three possible numbers of solutions for two linear equations in a system?

    $1$ solution, no solution, or infinitely many solutions. Lines can intersect once, never (parallel), or be the same line.

31. What is the solution to the system: $2x+y=9$ and $x-y=1$?

    $(\frac{10}{3}, \frac{7}{3})$. Add equations: $3x=10$, so $x=\frac{10}{3}$; then $y=\frac{7}{3}$.

32. What is the solution to the system: $3x+2y=12$ and $3x-2y=4$?

    $(\frac{8}{3},2)$. Add equations: $6x=16$, so $x=\frac{8}{3}$; then $y=2$.

33. What is the substitution method for solving a system used for?

    Replacing one variable using an equation to solve for the other variable. Substitute to reduce the system to one equation with one variable.

34. What is the solution to the system by inspection: $x+y=7$ and $x+y=7$?

    Infinitely many solutions (the same line). Identical equations describe the same line.

35. What should you do after finding one variable value in a system?

    Substitute it into an original equation to find the other variable. Use the known value to solve for the remaining unknown.

36. What condition on slopes and intercepts gives no solution for a linear system?

    Same slope, different $y$-intercepts (parallel distinct lines). Parallel lines never meet, so no point satisfies both.

37. What does it mean if a system has infinitely many solutions?

    The equations are equivalent and represent the same line. Every point on the line satisfies both equations.

38. Identify the solution by inspection: $y=2x+1$ and $y=2x-3$.

    No solution. Same slope (2) but different $y$-intercepts means parallel lines.

39. What does it mean if a system has no solution?

    The lines are parallel: same slope, different $y$-intercepts. Parallel lines never meet, so no $(x,y)$ satisfies both equations.

40. What is a quick inspection clue that a system has no solution?

    Same left side but different constants, like $x+y=2$ and $x+y=5$. Can't have same sum equal different values.

41. What is the solution to the system $3x-2y=4$ and $x-2y=0$?

    $(2,1)$. Subtract equations: $(3x-2y)-(x-2y)=4-0$, so $2x=4$, $x=2$.

42. What is the first step to solve by substitution if one equation is $y=mx+b$?

    Substitute $y=mx+b$ into the other equation. Since $y$ is already isolated, substitute directly.

43. What does it mean if two lines intersect at exactly one point on a graph?

    The system has exactly one solution. The lines cross at one point, giving a unique solution.

44. What is the solution to the system $y=2x+1$ and $y=9$?

    $(4,9)$. Substitute $y=9$ into first equation: $9=2x+1$, so $x=4$.

45. What is a quick inspection clue that a system has infinitely many solutions?

    One equation simplifies to the other, like $2x+2y=4$ and $x+y=2$. Equivalent equations represent the same line.

46. What is the first step to estimate a system solution by graphing?

    Graph both lines and approximate their intersection point $(x,y)$. Visual method to find where lines meet.

47. What does the solution of a system of two linear equations represent on a graph?

    The intersection point $(x,y)$ that satisfies both equations. Where two lines cross gives values that make both equations true.

48. What is the solution to the system $x+y=7$ and $x-y=1$?

    $(4,3)$. Add equations: $2x=8$, so $x=4$; subtract: $2y=6$, so $y=3$.

49. Solve by substitution: $y=x+1$ and $2x+y=10$.

    $(3,4)$. Substitute $y=x+1$ into $2x+y=10$: $2x+(x+1)=10$, so $3x=9$.

50. What is the solution type for $y=-\frac{1}{2}x+4$ and $2y=-x+8$?

    Infinitely many solutions (equivalent equations). Multiply first by $2$: $2y=-x+8$, matching the second.

51. Solve by inspection: $x+y=5$ and $x+y=5$.

    Infinitely many solutions. Identical equations represent the same line.

52. What is the solution to the system $x+y=5$ and $y=x+1$?

    $(2,3)$. Substitute second into first: $x+(x+1)=5$, so $x=2$, $y=3$.

53. After solving algebraically, what should you do to check the solution $(x,y)$?

    Substitute $(x,y)$ into both equations and verify both are true. Ensures your algebraic solution is correct.

54. Identify the number of solutions: $2x+4y=6$ and $x+2y=3$.

    Infinitely many solutions. Second equation is half the first; they're equivalent.

55. What does it mean if you get a true statement like $0=0$ after elimination?

    Infinitely many solutions. Both equations represent the same line.

56. What does it mean if two equations graph as the same line (equivalent equations)?

    The system has infinitely many solutions. Same line means every point on it satisfies both equations.

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