Explaining and Justifying Equation Solving Steps - Algebra 1

1. What property states that if $a=b$, then $a+c=b+c$ for any real $c$?

   Addition Property of Equality. This is the formal statement of adding the same value to both sides.

2. Identify the solution of $-2x+5=-1$.

   $x=3$. Subtract $5$, then divide by $-2$ to get $x = 3$.

3. What is the first recommended step to clear denominators in $\frac{x}{3}+2=5$?

   Multiply both sides by $3$. This eliminates the fraction by clearing the denominator.

4. Which property justifies rewriting $x\cdot \frac{1}{x}$ as $1$ when $x\ne 0$?

   Multiplicative Inverse Property. This property states that a nonzero number times its reciprocal equals one.

5. Identify the solution of $0.2x+1=3$.

   $x=10$. Subtract $1$, then divide by $0.2$ to get $x = 10$.

6. What conclusion is justified if solving leads to $0=0$?

   Infinitely many solutions. A true statement indicates the equation is an identity.

7. What restriction is required before multiplying an equation by $\frac{1}{x}$?

   $x \neq 0$. Multiplying by $\frac{1}{x}$ requires $x \neq 0$ since $\frac{1}{0}$ is undefined.

8. What restriction is required to use the Division Property of Equality on $\frac{a}{c}=\frac{b}{c}$?

   $c\ne 0$. Division by zero is undefined, so $c$ cannot equal zero.

9. Identify the solution of $-3x=12$.

   $x=-4$. Divide both sides by $-3$ to get $x = -4$.

10. What restriction is required before multiplying an equation by $\frac{1}{x}$?

    $x\ne 0$. Multiplying by $\frac{1}{x}$ requires $x \ne 0$ since $\frac{1}{0}$ is undefined.

11. What property states that if $a=b$, then $a$ may be replaced by $b$ in any expression?

    Substitution Property of Equality. This property allows replacing equal values in expressions.

12. What property justifies adding the same number to both sides of $a=b$ to get $a+c=b+c$?

    Addition Property of Equality. This property maintains equality when the same value is added to both sides.

13. Which step is valid to justify going from $x+8=3$ to $x=-5$?

    Subtract $8$ from both sides. This operation transforms the equation from $x + 8 = 3$ to $x = -5$.

14. Which property justifies rewriting $1\cdot x$ as $x$ during equation solving?

    Multiplicative Identity Property. This property states that multiplying by one doesn't change a value.

15. Identify the solution of $5(x-2)=15$.

    $x=5$. Divide by $5$, then add $2$ to get $x = 5$.

16. Identify the solution of $x+7=12$.

    $x=5$. Subtract $7$ from both sides to get $x = 5$.

17. What inverse operation undoes multiplying by $k$ in $kx$ when $k\ne 0$?

    Divide by $k$. Division is the inverse operation that undoes multiplication.

18. Which property justifies factoring $ax+ay$ as $a(x+y)$ when solving equations?

    Distributive Property. This property allows factoring out common terms.

19. Identify the solution of $\frac{x}{6}=-3$.

    $x=-18$. Multiply both sides by $6$ to get $x = -18$.

20. Which property justifies rewriting $x+3x$ as $4x$ while solving an equation?

    Combine like terms (distributive idea). Like terms have the same variable part and can be combined.

21. What property states that if $a=b$, then $a-c=b-c$ for any real $c$?

    Subtraction Property of Equality. This is the formal statement of subtracting the same value from both sides.

22. Identify the solution of $7-2x=1$.

    $x=3$. Subtract $7$, divide by $-2$ to get $x = 3$.

23. Find and correct the invalid step: from $x=3$ conclude $\frac{1}{x}=\frac{1}{3}$ without a condition.

    Valid only if $x\ne 0$. The reciprocal operation requires the denominator to be nonzero.

24. Which property justifies rewriting $x+0$ as $x$ during equation solving?

    Additive Identity Property. This property states that adding zero doesn't change a value.

25. Find and correct the invalid step: from $x=3$ conclude $\frac{1}{x}=\frac{1}{3}$ without a condition.

    Valid only if $x \ne 0$. The reciprocal operation requires the denominator to be nonzero.

26. Identify the solution of $\frac{x}{4}+3=5$.

    $x=8$. Subtract $3$, then multiply by $4$ to get $x = 8$.

27. Which property justifies rewriting $x+(-x)$ as $0$ during equation solving?

    Additive Inverse Property. This property states that a number plus its opposite equals zero.

28. Which property justifies going from $2x+6=10$ to $2x=4$?

    Subtraction Property of Equality. Subtracting $6$ from both sides removes the constant term.

29. Identify the solution of $3x-2=3x+4$.

    No solution. Subtracting $3x$ from both sides gives $-2 = 4$, which is false.

30. Identify the solution of $9=3x+0$.

    $x=3$. Divide both sides by $3$ to get $x = 3$.

31. Identify the property used to justify from $x-4=9$ to $x-4+4=9+4$.

    Addition Property of Equality. Adding $4$ to both sides removes the subtracted term.

32. What property justifies subtracting the same number from both sides of $a=b$ to get $a-c=b-c$?

    Subtraction Property of Equality. This property maintains equality when the same value is subtracted from both sides.

33. What is the standard goal when solving a linear equation in one variable?

    Isolate the variable (get $x$ alone). This creates an equation in the form $x = \text{value}$.

34. Which step correctly clears denominators in $\frac{x}{2}+\frac{x}{3}=5$?

    Multiply both sides by $6$. The LCD of $2$ and $3$ is $6$, so multiply by $6$.

35. Identify the property used to justify from $x+2=7$ to $x=5$.

    Subtraction Property of Equality. Subtracting $2$ from both sides isolates the variable.

36. Which operation is typically used first to solve $x-\frac{3}{2}=4$?

    Add $\frac{3}{2}$ to both sides. Adding the opposite of the subtracted term isolates $x$.

37. Identify the solution of $5x-1=2x+8$.

    $x=3$. Subtract $2x$ and add $1$, then divide by $3$ to get $x = 3$.

38. Identify the solution of $\frac{x}{2}+\frac{x}{3}=5$.

    $x=6$. After clearing denominators: $3x + 2x = 30$, so $x = 6$.

39. Identify the solution of $\frac{2x-1}{3}=5$.

    $x=8$. Multiply by $3$, add $1$, then divide by $2$ to get $x = 8$.

40. Identify the solution of $0.5x=6$.

    $x=12$. Divide both sides by $0.5$ to get $x = 12$.

41. Identify the solution of $2(x+4)=18$.

    $x=5$. Divide by $2$, then subtract $4$ to get $x = 5$.

42. What inverse operation undoes adding $k$ in $x+k$?

    Subtract $k$. Subtraction is the inverse operation that undoes addition.

43. Which step correctly clears denominators in $\frac{x}{2}+\frac{x}{3}=5$?

    Multiply both sides by $6$. The LCD of $2$ and $3$ is $6$, so multiply by $6$.

44. What conclusion is justified if solving leads to $0=7$?

    No solution. A false statement indicates the equation has no solution.

45. Which property justifies going from $2x=4$ to $x=2$?

    Division Property of Equality. Dividing by $2$ isolates the variable $x$.

46. What property states that if $a=b$, then $b=a$?

    Symmetric Property of Equality. This property allows switching the order of equal expressions.

47. Identify the solution of $3x+5=20$.

    $x=5$. Subtract $5$, then divide by $3$ to get $x = 5$.

48. What property states that if $a=b$ and $b=c$, then $a=c$?

    Transitive Property of Equality. This property chains equal expressions together.

49. What property justifies multiplying both sides of $a=b$ by $c$ to get $ac=bc$?

    Multiplication Property of Equality. This property maintains equality when both sides are multiplied by the same value.

50. Identify the solution set of $2(x+1)=2x+2$.

    Infinitely many solutions. Expanding shows $2x + 2 = 2x + 2$, which is always true.

51. What is the most direct way to check that $x=4$ solves $2x+1=9$?

    Substitute $x=4$ and verify $9=9$. Substitution confirms the solution by making both sides equal.

52. Identify the solution of $4x=28$.

    $x=7$. Divide both sides by $4$ to get $x = 7$.

53. Identify the solution of $3(x-1)+2=11$.

    $x=4$. Distribute, combine like terms, then solve to get $x = 4$.

54. What term describes an operation that reverses another operation when solving equations?

    Inverse operation. These operations cancel each other out to isolate variables.

55. Identify the solution of $2x+6=x+10$.

    $x=4$. Subtract $x$ and $6$, then solve to get $x = 4$.

56. Find and correct the invalid step: divide both sides of $x\cdot 0=5\cdot 0$ by $0$ to get $x=5$.

    Division by $0$ is invalid. Division by zero is undefined in mathematics.

57. Identify the property used to justify from $3(x+2)=21$ to $x+2=7$.

    Division Property of Equality. Dividing both sides by $3$ simplifies the parenthetical expression.

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