Solve Linear Equations With Rationals - 8th Grade Math

1. What is the solution to $\frac{1}{3}x - 2 = 4$?

   $x = 18$. Add 2, then multiply by 3: $(4+2) \cdot 3 = 18$.

2. What is the solution to $\frac{3}{4}x + \frac{1}{2} = 2$?

   $x = 2$. Subtract $\frac{1}{2}$, then multiply by $\frac{4}{3}$: $\frac{3}{2} \cdot \frac{4}{3} = 2$.

3. What is the solution of $\frac{3}{5}(x - 5) + 1 = 4$?

   $x = 10$. Distribute: $\frac{3}{5}x - 3 + 1 = 4$, so $\frac{3}{5}x = 6$.

4. What is the solution of $-3(x + 2) = 9$?

   $x = -5$. Distribute: $-3x - 6 = 9$, add 6, divide by -3.

5. What is $x$ in the equation $\frac{1}{2}(x+6)=5$?

   $x=4$. Multiply both sides by 2, then subtract 6.

6. What is the solution to $\frac{1}{2}(4x + 6) = 9$?

   $x = 3$. Distribute, then solve: $2x + 3 = 9$, so $2x = 6$, $x = 3$.

7. What is $x$ in the equation $\frac{2}{5}x-\frac{1}{5}=1$?

   $x=3$. Add $\frac{1}{5}$, then multiply by $\frac{5}{2}$.

8. What is $x$ in the equation $\frac{2}{3}(x-3)+1=3$?

   $x=6$. Distribute $\frac{2}{3}$, then isolate $x$.

9. What is the first step to solve $3(x-4)=15$?

   Distribute: $3x-12=15$. Apply $3(x-4)=3x-12$ to expand the left side.

10. What is the solution of $3x + 2 = x + 10$?

    $x = 4$. Subtract $x$ and 2: $2x = 8$, then divide by 2.

11. What is the solution of $2.5x + 0.5 = 3.0$?

    $x = 1$. Subtract 0.5 to get $2.5x = 2.5$, then divide by 2.5.

12. What is the distributive property for $a(b+c)$?

    $a(b+c)=ab+ac$. Multiply $a$ by each term inside the parentheses.

13. What is the solution of $\frac{x}{3} + \frac{x}{6} = 3$?

    $x = 6$. Combine: $\frac{2x}{6} + \frac{x}{6} = \frac{3x}{6} = \frac{x}{2} = 3$.

14. What is $x$ in the equation $2x+\frac{1}{2}=x+3$?

    $x=\frac{5}{2}$. Subtract $x$ and $\frac{1}{2}$ from both sides.

15. What does it mean to collect like terms in a linear equation?

    Combine terms with the same variable part (for example, $x$-terms). Simplifies by adding/subtracting coefficients of identical variables.

16. What is $x$ in the equation $\frac{3}{4}x=6$?

    $x=8$. Multiply both sides by $\frac{4}{3}$ to isolate $x$.

17. What is the solution of $\frac{1}{2}x + 3 = 7$?

    $x = 8$. Subtract 3, get $\frac{1}{2}x = 4$, then multiply by 2.

18. What is the solution to $x + \frac{3}{8} = \frac{7}{8}$?

    $x = \frac{1}{2}$. Subtract $\frac{3}{8}$ from both sides: $x = \frac{7}{8} - \frac{3}{8} = \frac{4}{8} = \frac{1}{2}$.

19. What is $x$ in the equation $x+\frac{2}{3}=\frac{5}{3}$?

    $x=1$. Subtract $\frac{2}{3}$ from both sides.

20. What is the solution to $\frac{2}{5}x = 6$?

    $x = 15$. Multiply both sides by $\frac{5}{2}$: $x = 6 \cdot \frac{5}{2} = 15$.

21. What is $x$ in the equation $\frac{3}{2}x+1=\frac{1}{2}x+5$?

    $x=4$. Subtract $\frac{1}{2}x$ and 1 from both sides.

22. What is $x$ in the equation $5-2(x+1)=1$?

    $x=1$. Distribute $-2$, then solve for $x$.

23. What is $x$ in the equation $4x-3=2x+5$?

    $x=4$. Subtract $2x$ and add 3 to both sides.

24. What is the solution to $\frac{2}{3}(x - 6) = 4$?

    $x = 12$. Multiply by $\frac{3}{2}$, then add 6: $4 \cdot \frac{3}{2} + 6 = 12$.

25. What does it mean to solve a linear equation in one variable?

    Find the value of $x$ that makes the equation true. The solution is the value that satisfies the equation.

26. What is the solution of $\frac{1}{3}x + 2 = \frac{2}{3}x - 1$?

    $x = 9$. Subtract $\frac{1}{3}x$ and add 1: $\frac{1}{3}x = 3$.

27. What is the solution to $\frac{1}{2}x - \frac{3}{4} = \frac{1}{4}x + \frac{1}{2}$?

    $x = 5$. Subtract $\frac{1}{4}x$, add $\frac{3}{4}$: $\frac{1}{4}x = \frac{5}{4}$, so $x = 5$.

28. What is the solution of $2(x - 5) = 6$?

    $x = 8$. Distribute: $2x - 10 = 6$, add 10, then divide by 2.

29. What is the solution of $4(x - 1) - 2x = 10$?

    $x = 7$. Distribute and combine: $4x - 4 - 2x = 10$, so $2x = 14$.

30. What is the distributive property for $a(b-c)$?

    $a(b-c)=ab-ac$. Multiply $a$ by each term, keeping the subtraction.

31. What is the inverse operation for subtracting $\frac{7}{3}$ from $x$ in $x - \frac{7}{3}$?

    Add $\frac{7}{3}$. Addition undoes subtraction to isolate the variable.

32. What is the solution of $\frac{2}{3}(x + 6) = 10$?

    $x = 9$. Multiply by $\frac{3}{2}$: $x + 6 = 15$, then subtract 6.

33. What is the solution to $\frac{5}{6}x - \frac{1}{3} = \frac{1}{2}$?

    $x = 1$. Add $\frac{1}{3}$ to get $\frac{5}{6}x = \frac{5}{6}$, then $x = 1$.

34. What is the solution to $\frac{1}{4}x + \frac{1}{2}x = 6$?

    $x = 8$. Combine like terms: $\frac{3}{4}x = 6$, then $x = 8$.

35. What is the rule for eliminating a fraction coefficient like $\frac{3}{4}x$ in an equation?

    Multiply both sides by the least common denominator (LCD). This clears fractions by making all coefficients whole numbers.

36. What is the solution of $\frac{3}{4}x = 6$?

    $x = 8$. Multiply both sides by $\frac{4}{3}$ to isolate $x$.

37. What is the solution to $2(x - \frac{3}{2}) = 5$?

    $x = 4$. Distribute 2, then solve: $2x - 3 = 5$, so $x = 4$.

38. What is the solution to $\frac{3}{5}x + 2 = \frac{1}{5}x + 6$?

    $x = 10$. Subtract $\frac{1}{5}x$ and 2: $\frac{2}{5}x = 4$, so $x = 10$.

39. What is $x$ in the equation $\frac{1}{3}x+\frac{1}{2}=2$?

    $x=\frac{9}{2}$. Subtract $\frac{1}{2}$, then multiply by 3.

40. What does it mean to collect like terms in $2x + 5 - x$?

    Combine terms with the same variable and exponent. Add or subtract coefficients of matching variables.

41. What is the distributive property for $k(a - b)$?

    $k(a - b) = ka - kb$. Distribute $k$ to both terms, keeping the subtraction sign.

42. Identify the simplified expression: $\frac{1}{2}(6x - 8)$.

    $3x - 4$. Distribute $\frac{1}{2}$ to both terms: $\frac{6x}{2} - \frac{8}{2}$.

43. What is $x$ in the equation $3(x-4)=15$?

    $x=9$. Distribute, add 12, then divide by 3.

44. What is the first step to solve $ax + b = c$ for $x$ when $a \ne 0$?

    Subtract $b$ from both sides to get $ax = c - b$. Isolating the term with $x$ is the first step in solving for $x$.

45. What is the result after combining like terms: $\frac{3}{4}x - \frac{1}{2}x$?

    $\frac{1}{4}x$. Convert to common denominator: $\frac{3}{4}x - \frac{2}{4}x = \frac{1}{4}x$.

46. What property lets you rewrite $3(x - 4)$ as $3x - 12$?

    Distributive property. Multiply each term inside parentheses by the outside factor.

47. What is the solution to $\frac{x}{3} + \frac{x}{6} = 5$?

    $x = 10$. Combine fractions: $\frac{2x}{6} + \frac{x}{6} = \frac{3x}{6} = \frac{x}{2} = 5$.

48. What is the LCD of $\frac{1}{6}$, $\frac{1}{4}$, and $\frac{1}{3}$?

    $12$. Find the smallest number divisible by 6, 4, and 3.

49. What is the solution of $\frac{1}{2}(x - 4) = \frac{3}{2}$?

    $x = 7$. Multiply by 2: $x - 4 = 3$, then add 4.

50. What is the solution to $\frac{2}{3}x + \frac{1}{2} = \frac{1}{6}x + \frac{5}{2}$?

    $x = 4$. Subtract $\frac{1}{6}x$ and $\frac{1}{2}$: $\frac{1}{2}x = 2$, so $x = 4$.

51. What is the distributive property for $k(a + b)$?

    $k(a + b) = ka + kb$. Multiply $k$ by each term inside the parentheses.

52. What is the solution of $0.5x + 1.5 = 5.5$?

    $x = 8$. Subtract 1.5 to get $0.5x = 4$, then divide by 0.5.

53. What is $x$ in the equation $3x-2(x+4)=1$?

    $x=9$. Distribute, combine like terms, then solve.

54. What is $x$ in the equation $x-\frac{5}{6}=\frac{1}{6}$?

    $x=1$. Add $\frac{5}{6}$ to both sides to get $x=\frac{6}{6}=1$.

55. What is the solution of $\frac{1}{2}x - \frac{3}{4} = \frac{5}{4}$?

    $x = 4$. Add $\frac{3}{4}$ to get $\frac{1}{2}x = 2$, then multiply by 2.

56. What property lets you multiply both sides of $a=b$ by the same nonzero number $c$?

    Multiplication Property of Equality. If $a=b$, then $ac=bc$ for any nonzero $c$.

57. What property lets you add the same number to both sides of $a=b$?

    Addition Property of Equality. If $a=b$, then $a+c=b+c$ for any number $c$.

58. What is the solution of $1.2x - 0.4 = 2.0$?

    $x = 2$. Add 0.4 to get $1.2x = 2.4$, then divide by 1.2.

59. What is $x$ in the equation $\frac{1}{2}x=7$?

    $x=14$. Multiply both sides by 2 to isolate $x$.

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