Solve Real-World System Problems - 8th Grade Math

1. What is the meaning of the solution to a system on a coordinate plane?

   The intersection point of the two lines

2. Identify the solution to the system: $2x+3y=12$ and $x+3y=9$.

   $(3,2)$. Subtract second from first: $x=3$; substitute to get $y=2$.

3. Find the miles $m$ if the system is $C=3+2m$ and $C=19$.

   $m=8$. Substitute: $19=3+2m$, so $2m=16$, thus $m=8$ miles.

4. Identify the solution to the system: $2x+y=9$ and $x+y=6$.

   $(3,3)$. Subtract second from first: $x=3$; substitute to get $y=3$.

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

   The lines intersect at exactly one point. The unique intersection point is the solution.

6. What is the substitution method for solving a system of linear equations?

   Solve one equation for a variable, substitute into the other. This reduces the system to one equation with one variable.

7. Identify whether the system has $0,1,$ or infinitely many solutions: $2x+4y=8$ and $x+2y=4$.

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

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

   One solution. Different slopes ($m=-1$ and $m=1$) guarantee intersection.

9. What system models: You buy $x$ pens at $2 each and $y$ notebooks at $5 each for $19, total $6$ items?

   $x+y=6$ and $2x+5y=19$. First equation: total items; second: total cost at given prices.

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

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

11. Identify the solution to the system: $y=2x+1$ and $y=9-x$.

    $x=\frac{8}{3},\ y=\frac{19}{3}$. Set equal: $2x+1=9-x$, so $3x=8$, $x=\frac{8}{3}$; then $y=\frac{19}{3}$.

12. What is the solution type for the system: $y=3x+2$ and $y=3x-5$?

    No solution. Parallel lines (same slope $m=3$) with different y-intercepts.

13. What is the substitution goal when solving a system of linear equations?

    Rewrite one equation as a variable expression and substitute. Replace one variable with an expression to solve for the other.

14. What does it mean if a system has infinitely many solutions in terms of the lines on a graph?

    The lines are the same line (coincident). Every point on one line is also on the other line.

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

    The equations represent the same line. Both equations simplify to the same slope and y-intercept.

16. Identify the system for: "$3$ adult tickets and $2$ child tickets cost $46$; adult is $4$ more."

    $3a+2c=46$ and $a=c+4$

17. What is the elimination method for solving a system of linear equations?

    Add or subtract equations to eliminate one variable. Combining equations strategically simplifies to one variable.

18. Identify the solution to the system: $x+2y=10$ and $x-y=1$.

    $(4,3)$. From second equation $x=y+1$; substitute to get $y=3$, $x=4$.

19. What does it mean for an ordered pair $(x,y)$ to be a solution to a system of two linear equations?

    $(x,y)$ makes both equations true when substituted. Both equations must be satisfied simultaneously by the same values.

20. What system models: $x$ adult tickets and $y$ student tickets, $x+y=30$, adults $8$ dollars, students $5$ dollars, total $201$ dollars?

    $x+y=30$ and $8x+5y=201$. First equation: total tickets; second: total revenue.

21. Identify whether the system has $0,1,$ or infinitely many solutions: $y=-x+5$ and $y=2x-1$.

    $1$ solution. Different slopes (-1 and 2) guarantee one intersection.

22. What does it mean for an ordered pair $(x, y)$ to solve a system of two equations?

    $(x, y)$ makes both equations true at the same time

23. What system models: perimeter $50$ of a rectangle with length $x$ and width $y$, and $x=y+5$?

    $2x+2y=50$ and $x=y+5$. Perimeter formula and length-width relationship.

24. What does it mean if a system has exactly one solution in terms of the lines on a graph?

    The lines intersect at exactly one point. A unique solution occurs where two non-parallel lines meet.

25. Solve the system: $y=x-2$ and $y=-2x+7$.

    $(3,1)$

26. Solve the system: $3x+2y=12$ and $x+2y=4$.

    $(4,0)$

27. What condition on slopes indicates a system has no solution (inconsistent)?

    Same slope, different $y$-intercepts

28. What is the first step when translating a word problem into a system of equations?

    Define variables for the two unknown quantities

29. Identify the solution to the system: $3x+2y=12$ and $x+2y=8$.

    $(2,3)$. Subtract equations: $2x=4$, so $x=2$; then $2+2y=8$, so $y=3$.

30. What does it mean for an ordered pair $(x,y)$ to be a solution to a system of equations?

    $(x,y)$ makes both equations true when substituted. Both equations must be satisfied simultaneously.

31. What is the first step to solve a word problem with a system of equations?

    Define variables for the unknown quantities. Variables represent what you're solving for.

32. Solve the system by substitution: $y=2x+1$ and $x+y=10$.

    $(3,7)$

33. Solve the system: $2x+y=11$ and $x-y=1$.

    $(4,3)$

34. What is the solution type for the system: $2x+4y=10$ and $x+2y=5$?

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

35. Identify the system for: "Two numbers add to $20$ and their difference is $4$."

    $x+y=20$ and $x-y=4$

36. Identify the solution to the system: $3x-2y=4$ and $x-y=2$.

    $(0,-2)$. From second equation $x=y+2$; substitute to get $y=-2$, $x=0$.

37. What condition indicates a system has infinitely many solutions (dependent)?

    Equations are equivalent (same line)

38. What does a negative solution like $(x,y)=(-2,5)$ indicate in a real-world context?

    Check context; it may be invalid if quantities cannot be negative

39. Solve the system by elimination: $x+y=9$ and $x-y=1$.

    $(5,4)$

40. 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.

41. What are the three possible numbers of solutions for two linear equations in two variables?

    $0$ solutions, $1$ solution, or infinitely many solutions

42. What does it mean if a system has no solution in terms of the lines on a graph?

    The lines are parallel and never intersect. Same slopes but different y-intercepts mean no intersection.

43. What are the ticket prices if $2a+3s=27$ and $a+2s=16$?

    $a=6,\ s=5$. From second: $a=16-2s$; substitute and solve to get prices.

44. Which form is most useful to graph quickly: standard form $Ax+By=C$ or slope-intercept $y=mx+b$?

    Slope-intercept form $y=mx+b$. Shows slope and y-intercept directly for easy graphing.

45. What system models: 2 adult tickets and 3 student tickets cost $27, and 1 adult and 2 student cost $16?

    $2a+3s=27$ and $a+2s=16$. Let $a$ = adult price, $s$ = student price; translate costs to equations.

46. Identify the system for: "A rectangle has perimeter $54$ and length is $3$ more than width."

    $2l+2w=54$ and $l=w+3$

47. Identify whether the system has $0,1,$ or infinitely many solutions: $y=2x+3$ and $y=2x-1$.

    $0$ solutions. Same slope (2) but different y-intercepts means parallel lines.

48. What system models: A taxi charges a $3 fee plus $2 per mile; a ride costs $19 for $m$ miles?

    $C=3+2m$ and $C=19$. First equation: cost formula; second: actual cost for the ride.

49. Identify the solution to the system: $x+y=10$ and $x-y=2$.

    $(6,4)$. Add equations: $2x=12$, so $x=6$; subtract: $2y=8$, so $y=4$.

50. What is the elimination goal when solving a system by adding or subtracting equations?

    Make one variable cancel to eliminate it. Add/subtract equations to get one variable with coefficient 0.

51. Identify the intersection point of $y=4$ and $x+2y=14$.

    $(6,4)$. Substitute $y=4$ into second equation: $x+8=14$, so $x=6$.

52. After solving a real-world system, what is the required final step to confirm correctness?

    Substitute the solution into both original equations

53. Identify the meaning of the solution $(x,y)$ in a cost problem with equations for total items and total cost.

    $(x,y)$ gives the quantities of the two item types. Solution tells how many of each item type was purchased.

54. Identify the solution type: $3x-6y=9$ and $x-2y=3$.

    Infinitely many solutions

55. Identify the solution type: $y=2x+3$ and $y=2x-5$.

    No solution

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