Understand System Solutions as Intersections - 8th Grade Math

1. Which option is true for $y=-3x+6$ and $2y=-6x+12$: one solution, none, or infinitely many?

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

2. Identify whether $(1,2)$ is the intersection of $y=2x$ and $y=x+1$.

   Yes, $(1,2)$ satisfies both equations. Check: $2=2(1)$ ✓ and $2=1+1$ ✓.

3. What must be true about a point $(x,y)$ for it to be the intersection of two lines in a system?

   It satisfies both equations when substituted. Intersection points make both equations true.

4. What is the number of solutions to the system $y=3x-1$ and $y=3x+2$?

   No solution. Same slope, different intercepts = parallel.

5. Which statement matches infinitely many solutions: same slope with same $y$-intercept or same slope with different $y$-intercepts?

   Same slope with same $y$-intercept. Same line means all points work.

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

   The lines are parallel and never intersect. Same slope but different $y$-intercepts prevent crossing.

7. Which option is true for $y=2x+3$ and $y=2x-1$: one solution, none, or infinitely many?

   No solution. Same slope $m=2$, different intercepts mean parallel.

8. What is the solution of the system $y=x-5$ and $y=3x-9$?

   $(2,-3)$. Set equal: $x-5=3x-9$, giving $x=2$, then $y=-3$.

9. What does a solution $(x,y)$ to a system of two linear equations represent on their graphs?

   The intersection point of the two lines at $(x,y)$. Where both lines meet, satisfying both equations simultaneously.

10. What is the meaning of the ordered pair $(x,y)$ that satisfies both equations in a system?

    $(x,y)$ makes both equations true at the same time. The point lies on both lines simultaneously.

11. Identify the graphical meaning of the $y$-coordinate of a system's solution point.

    The shared $y$-value where the two lines intersect. Both lines reach this height at intersection.

12. Which option describes the solution set if two lines intersect at $(3,-2)$?

    The system solution is $(3,-2)$. The intersection point is the only solution.

13. What is the solution to the system if the lines intersect at $(-1,4)$?

    $(-1,4)$. The intersection point is the solution.

14. What is the solution of the system $y=2x+1$ and $y=-x+4$?

    $(1,3)$. Set equations equal: $2x+1=-x+4$, solve for $x=1$.

15. What does it mean if two lines are parallel and never intersect on a coordinate plane?

    The system has no solution. Parallel lines never meet, so no common points.

16. Which statement matches a system with one solution: the lines have the same slope or different slopes?

    Different slopes. Lines with different slopes must intersect once.

17. What is the number of solutions to the system $y=-2x+7$ and $y=-2x+7$?

    Infinitely many solutions. Identical equations = same line.

18. What is the solution of the system $x+y=5$ and $x-y=1$?

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

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

    The lines are the same line and overlap completely. Same slope and $y$-intercept make them identical.

20. What does it mean if two lines intersect at exactly one point on a coordinate plane?

    The system has exactly one solution. Lines that cross once have one common point.

21. Which graph feature shows the solution to a linear system: the $y$-intercept or the intersection point?

    The intersection point. Solutions appear where graphs meet, not at axes.

22. Identify the solution set if the two equations graph as the same line.

    Infinitely many solutions (all points on the line). Same line means every point is a solution.

23. Which point is the solution to the system $y=2x$ and $y=x+3$?

    $(3,6)$. Solve: $2x=x+3$ gives $x=3$, then $y=6$.

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

    Infinitely many solutions. Second equation is $2×$ the first equation.

25. Which relationship between equations indicates infinitely many solutions?

    One equation is a multiple of the other. Equivalent equations represent the same line.

26. Which statement matches a system with no solution: same slope with different $y$-intercepts or different slopes?

    Same slope with different $y$-intercepts. Parallel lines never meet.

27. What does a solution $(x,y)$ to a system of two linear equations represent on the graphs?

    The intersection point of the two lines at $(x,y)$. Where both lines meet, both equations are satisfied.

28. Identify the number of solutions if the graphs intersect at $(3,-2)$.

    One solution. Lines crossing at one point means one solution.

29. Identify the graphical meaning of the $x$-coordinate of a system's solution point.

    The shared $x$-value where the two lines intersect. Both lines pass through this $x$-position.

30. Which relationship between slopes indicates two lines might have exactly one solution?

    Different slopes. Non-parallel lines must cross at one point.

31. Identify whether $(2,5)$ is a solution to $y=2x+1$ and $y=-x+7$.

    Yes, $(2,5)$ satisfies both equations. Check: $5=2(2)+1$ ✓ and $5=-2+7$ ✓.

32. What is the intersection point of $x=4$ and $y=-3$?

    $(4,-3)$. Vertical and horizontal lines meet at one point.

33. Find the intersection point of $y=1$ and $x=-2$.

    $(-2,1)$. Horizontal and vertical lines meet at one point.

34. Which point is the solution to the system $y=x+1$ and $y=-x+5$?

    $(2,3)$. Solve: $x+1=-x+5$ gives $x=2$, then $y=3$.

35. Identify the number of solutions for $3x-2y=8$ and $6x-4y=10$.

    No solution. Second isn't a multiple of first; parallel lines.

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

    The lines intersect at exactly one point. Two distinct lines cross at a single shared point.

37. What is the solution of the system $2x+y=7$ and $x-y=1$?

    $( \,\frac{8}{3},\frac{5}{3}\,)$. Add equations: $3x=8$, so $x=\frac{8}{3}$, then $y=\frac{5}{3}$.

38. Identify the number of solutions if the graphs are parallel distinct lines.

    No solution. Parallel lines never intersect.

39. Which relationship between slopes indicates two lines might have no solution?

    Equal slopes with different $y$-intercepts. Parallel lines never meet when slopes match.

40. What does it mean if two lines lie on top of each other (coincide) on a coordinate plane?

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

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