Equations With One Variable - PSAT Math

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

   $x=7$. Multiply by $5$ to get $2x+1=15$, then solve for $x$.

2. What is the solution set of $3(x-2)=3x-6$?

   Infinitely many solutions. Expanding gives $3x-6=3x-6$, always true for any $x$.

3. What property justifies adding the same number to both sides of $x-7=12$?

   Addition Property of Equality. Allows adding the same value to both sides to maintain equality.

4. What is the result type if solving simplifies to $0=5$?

   No solution. A false statement means no value satisfies the equation.

5. Identify the solution to $\frac{x}{5}+2=9$.

   $x=35$. Subtract 2: $\frac{x}{5}=7$, then multiply by 5.

6. What is the first step to solve $\frac{x}{3}+\frac{x}{2}=5$ efficiently?

   Multiply both sides by $6$. Eliminates fractions by finding LCD of 3 and 2.

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

   $x=7$. Multiply by 5, subtract 1, divide by 2: $14÷2=7$.

8. Identify the solution to $2x+5=3x-1$.

   $x=6$. Subtract $2x$ and add 1 to get $x=6$.

9. What is the solution to $x+8=19$?

   $x=11$. Subtract $8$ from both sides to isolate $x$.

10. What property justifies multiplying both sides of $\frac{x}{5}=9$ by $5$?

    Multiplication Property of Equality. Allows multiplying both sides by the same non-zero value.

11. What is the solution to $4(x-2)=20$?

    $x=7$. Divide by $4$ to get $x-2=5$, then add $2$: $x = 7$.

12. What does it mean for two expressions to be equivalent for all $x$?

    They have the same value for every $x$ in the domain. Equivalent expressions produce identical outputs for all inputs.

13. What is the solution to $x^2=49$?

    $x=7$ or $x=-7$. Take square root of both sides: $x=±7$.

14. What is the solution to the equation $\frac{x}{4}=9$?

    $x=36$. Multiply both sides by 4: $x = 9 × 4 = 36$.

15. What is the result of dividing both sides of an equation by $0$?

    It is undefined and not allowed. Division by zero is mathematically undefined.

16. What property justifies multiplying both sides of an equation by the same nonzero number?

    Multiplication Property of Equality. Allows multiplying both sides by same nonzero value to maintain equality.

17. What is the solution of $2x + 3 = 2x - 5$?

    No solution. Same variable terms cancel, leaving $3 = -5$, which is false.

18. What is the solution to $2x+5=2x-1$?

    No solution. Same $x$ coefficient on both sides with different constants.

19. What is the correct classification if an equation simplifies to $0=0$?

    Infinitely many solutions. A true statement like $0=0$ means all values work.

20. Which value is excluded from the domain of $\frac{1}{x-4}$ when solving equations?

    $x\ne 4$. Division by zero occurs when denominator equals zero: $x - 4 = 0$.

21. What is the solution to $\frac{3}{4}x=18$?

    $x=24$. Multiply by $\frac{4}{3}$: $x = 18 × \frac{4}{3} = 24$.

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

    $x=14$. Multiply both sides by 3, then add 2: $12+2=14$.

23. What is the solution set of $|x-3|=5$?

    $x=8$ or $x=-2$. Absolute value equation: $x - 3 = 5$ or $x - 3 = -5$.

24. What is the solution of $0.5x + 3 = 11$?

    $x = 16$. Subtract 3, then divide by 0.5: $8 ÷ 0.5 = 16$.

25. What is the solution to $\frac{x}{6}=4$?

    $x=24$. Multiply both sides by 6: $4×6=24$.

26. What is the correct classification if an equation simplifies to $0=5$?

    No solution. A false statement like $0=5$ means no value works.

27. What is the distributive property written with a variable?

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

28. What is the solution of $x^2 = 49$?

    $x = 7$ or $x = -7$. Take square root of both sides: $sqrt{49} = ±7$.

29. What is the solution to $3(x-1)+2=2x+5$?

    $x=6$. Expand: $3x - 3 + 2 = 2x + 5$, simplify to $3x - 1 = 2x + 5$, so $x = 6$.

30. What is the solution to $3(x-1)=3x-3$?

    All real numbers. Distributing gives $3x-3=3x-3$, an identity true for all $x$.

31. What is the solution to $7-3x=1$?

    $x=2$. Subtract $7$, divide by $-3$: $-3x = -6$, so $x = 2$.

32. What is the solution to $\frac{x}{6}=-4$?

    $x=-24$. Multiply both sides by $6$ to isolate $x$.

33. What is $x$ if $\frac{x}{5}-3=1$?

    $x=20$. Add $3$ to get $\frac{x}{5}=4$, then multiply by $5$.

34. Identify the solution set of $3(x-2)=3x+1$.

    No solution. Expands to $3x-6=3x+1$, simplifies to $-6=1$.

35. What property justifies adding the same number to both sides of $a=b$?

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

36. What is the solution to the equation $2(x+3)=18$?

    $x=6$. Divide by 2, then subtract 3: $x + 3 = 9$, so $x = 6$.

37. What is the solution to $x+9=2$?

    $x=-7$. Subtract $9$ from both sides: $x = 2 - 9 = -7$.

38. What does it mean for $x=a$ to be a solution of an equation in one variable?

    $x=a$ makes both sides equal when substituted. Substituting $a$ for $x$ makes the equation true.

39. What is the solution to the equation $\frac{x-5}{3}=4$?

    $x=17$. Multiply by 3, then add 5: $x - 5 = 12$, so $x = 17$.

40. What is the solution to the equation $12-3(x-2)=6$?

    $x=4$. Distribute: $12 - 3x + 6 = 6$, so $18 - 3x = 6$, thus $x = 4$.

41. What is the solution to $2x+4=2x-1$?

    No solution. Subtracting $2x$ gives $4=-1$, which is false.

42. What is the solution to $x+7=19$?

    $x=12$. Subtract 7 from both sides: $19-7=12$.

43. What is the solution of $3x - 5 = 16$?

    $x = 7$. Add 5 to both sides, then divide by 3: $21 ÷ 3 = 7$.

44. What conclusion follows if solving gives $0=5$?

    No solution. A false statement means the equation has no valid solutions.

45. What is the solution to $5x=35$?

    $x=7$. Divide both sides by $5$ to isolate $x$.

46. What is the inverse operation of multiplying by $k$ (with $k\ne 0$), as in $kx$?

    Divide by $k$. Division undoes multiplication when $k\ne 0$.

47. What is the solution to $0.5x+2=7$?

    $x=10$. Subtract 2: $0.5x=5$, then divide by 0.5 to get $x=10$.

48. What is the solution of $x^2 - 9 = 0$?

    $x = 3$ or $x = -3$. Factor: $(x + 3)(x - 3) = 0$, so $x = ±3$.

49. What is the solution of $\frac{x}{4}=9$?

    $x=36$. Multiply both sides by $4$: $x=9×4$.

50. What is the solution to $0.5x+1=4$?

    $x=6$. Subtract 1, then multiply by 2: $0.5x = 3$, so $x = 6$.

51. What is the solution of $3x-4=11$?

    $x=5$. Add $4$ to both sides, then divide by $3$.

52. What is $x$ if $0.2x=6$?

    $x=30$. Divide both sides by $0.2$: $x = 6 ÷ 0.2 = 30$.

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

    Infinitely many solutions. Expands to $2x+2=2x+2$, always true.

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

    $x=8$. Multiply by 3: $2x-1=15$, add 1, divide by 2.

55. What is the solution of $4x+3=2x+15$?

    $x=6$. Subtract $2x$ and $3$ from both sides: $2x=12$.

56. What is the solution to $2(x+5)=18$?

    $x=4$. Divide by 2, then subtract 5: $9-5=4$.

57. What is the solution to $6-3(x-2)=0$?

    $x=4$. Distribute: $6-3x+6=0$, so $12-3x=0$, thus $x=4$.

58. What is the solution to $x+8=19$?

    $x=11$. Subtract 8 from both sides: $19-8=11$.

59. What operation property justifies multiplying both sides of $a=b$ by the same nonzero number?

    Multiplication Property of Equality. States that if $a=b$, then $ac=bc$ for any nonzero number $c$.

60. What is the solution to the equation $(x-3)^2=16$?

    $x=7$ or $x=-1$. Take square root: $x - 3 = ±4$, so $x = 7$ or $x = -1$.

61. What is the solution to the equation $\frac{3}{x}=6$?

    $x=\frac{1}{2}$. Cross multiply: $3 = 6x$, so $x = \frac{3}{6} = \frac{1}{2}$.

62. What is the solution to $|2x+1|=7$?

    $x=3$ or $x=-4$. The expression $2x+1$ equals $7$ or $-7$.

63. What is the rule for clearing a denominator in $\frac{x}{5}=7$?

    Multiply both sides by $5$. Eliminates the denominator to isolate $x$.

64. What is the solution of $2(x+3)=2x+5$?

    No solution. Distributing gives $2x+6=2x+5$, which simplifies to $6=5$.

65. What is the solution to $|x-3|=5$?

    $x=8$ or $x=-2$. The expression $x-3$ equals $5$ or $-5$.

66. What is $x$ if $5x=35$?

    $x=7$. Divide both sides by $5$: $x = 35 ÷ 5 = 7$.

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

    $x=7$. Multiply by 5: $2x+1=15$, so $2x=14$, thus $x=7$.

68. What is the solution of $\frac{x}{4} = 9$?

    $x = 36$. Multiply both sides by 4: $9 × 4 = 36$.

69. What is the solution to $4x-3=2x+9$?

    $x=6$. Subtract $2x$ from both sides: $2x - 3 = 9$, so $x = 6$.

70. What is the solution to $x-3=2x+5$?

    $x=-8$. Subtract $x$ and add 3: $-3=x+5$, so $x=-8$.

71. What is the solution to $5x=35$?

    $x=7$. Divide both sides by $5$: $x = 35 ÷ 5 = 7$.

72. What is the solution set of $3(x-1)=3x+2$?

    No solution. Expanding gives $3x - 3 = 3x + 2$, which simplifies to $-3 = 2$ (false).

73. What is the solution set of $4(x-1)=4x-4$?

    Infinitely many solutions. Distributing gives $4x-4=4x-4$, which is always true.

74. What is $x$ if $0.2x+1=3$?

    $x=10$. Subtract $1$ to get $0.2x=2$, divide by $0.2$.

75. What is the solution to $0.2x=6$?

    $x=30$. Divide both sides by $0.2$: $x = 6 ÷ 0.2 = 30$.

76. What is $x$ if $\frac{x}{4}=-3$?

    $x=-12$. Multiply both sides by $4$: $x = -3 × 4 = -12$.

77. What is the solution to $4x+3=2x+15$?

    $x=6$. Subtract $2x$ and $3$ from both sides, then divide by $2$.

78. What is the solution of $7-2x=1$?

    $x=3$. Subtract $7$: $-2x=-6$, then divide by $-2$.

79. What is the solution to $7-2x=1$?

    $x=3$. Subtract 7, then divide by $-2$: $-6÷(-2)=3$.

80. What is the solution to $0.2x=8$?

    $x=40$. Divide both sides by 0.2: $8÷0.2=40$.

81. What property justifies adding the same number to both sides of $a=b$?

    Addition property of equality. This property maintains equality when adding to both sides.

82. What is the solution to the equation $3x-5=16$?

    $x=7$. Add 5 to both sides, then divide by 3: $3x = 21$, so $x = 7$.

83. What is the solution to the equation $4x+6=2x+18$?

    $x=6$. Subtract $2x$ and 6 from both sides: $2x = 12$, so $x = 6$.

84. What is the solution to $2x+5=17$?

    $x=6$. Subtract $5$, then divide by $2$: $2x = 12$, so $x = 6$.

85. What is the solution set of $3(x-1)=3x+2$?

    No solution. Distributing gives $3x-3=3x+2$, which simplifies to $-3=2$.

86. What is the solution of $5x = 35$?

    $x = 7$. Divide both sides by 5: $35 ÷ 5 = 7$.

87. What is $x$ if $\frac{2x+1}{5}=3$?

    $x=7$. Multiply by $5$ to get $2x+1=15$, then solve.

88. What is the solution set of $2(x+1)=2x+2$?

    All real numbers. Expanding gives $2x + 2 = 2x + 2$, which is always true.

89. What is $x$ if $|x-3|=7$?

    $x=10$ or $x=-4$. The absolute value equation gives $x-3=7$ or $x-3=-7$.

90. What conclusion follows if solving gives $0=0$?

    Infinitely many solutions. A true statement means all values of $x$ satisfy the equation.

91. What is the solution set of $4(x-1)=4x-4$?

    Infinitely many solutions. Expanding gives $4x-4=4x-4$, which is always true.

92. What is the solution to $0.5x=8$?

    $x=16$. Divide both sides by 0.5: $8÷0.5=16$.

93. What is the solution to $|x-3|=5$?

    $x=8$ or $x=-2$. $x-3=5$ or $x-3=-5$, giving two solutions.

94. What property justifies adding the same number to both sides of $x-7=12$?

    Addition Property of Equality. Allows adding the same value to both sides to maintain equality.

95. What is the solution to $x^2=49$?

    $x=7$ or $x=-7$. Take square root of both sides: $\sqrt{49}=±7$.

96. What is the solution of $5x=35$?

    $x=7$. Divide both sides by $5$: $35÷5=7$.

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

    $x=-12$. Multiply both sides by 4: $-3×4=-12$.

98. What is the solution to $2(x+3)=x+11$?

    $x=5$. Expand left side: $2x + 6 = x + 11$, then $x = 5$.

99. What is $x$ if $5-2(3x-1)=1$?

    $x=1$. Distribute: $5-6x+2=1$, then $7-6x=1$, so $x=1$.

100. What is the solution of $|x-3|=5$?

     $x=8$ or $x=-2$. $x-3=5$ or $x-3=-5$, giving two solutions.

101. What is the inverse operation of adding $k$ to a variable, as in $x+k$?

     Subtract $k$. Subtraction undoes addition.

102. What is the solution to $x^2-9=0$?

     $x=3$ or $x=-3$. Factor: $(x - 3)(x + 3) = 0$, so $x = 3$ or $x = -3$.

103. What is the solution of $3(x - 2) + 1 = 10$?

     $x = 5$. Distribute, combine: $3x - 6 + 1 = 10$, so $3x = 15$, $x = 5$.

104. Identify the solution type if simplifying gives $0x=9$.

     No solution. Since $0$ times any $x$ cannot equal $9$.

105. What is the solution to $3(x+2)=18$?

     $x=4$. Divide by $3$ to get $x+2=6$, then subtract $2$.

106. What property justifies adding the same number to both sides of $a=b$?

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

107. What is the solution to $7-2x=1$?

     $x=3$. Subtract 7, then divide by -2: $-2x = -6$, so $x = 3$.

108. What is the solution of $\frac{2x + 1}{5} = 3$?

     $x = 7$. Multiply by 5, subtract 1, divide by 2: $14 ÷ 2 = 7$.

109. Identify the solution to $5(x+2)-3x=14$.

     $x=2$. Distribute: $5x+10-3x=14$, simplify to $2x+10=14$.

110. What is the solution to $3x+5=20$?

     $x=5$. Subtract 5, then divide by 3: $3x = 15$, so $x = 5$.

111. What is the solution set of $2(x+1)=2x+2$?

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

112. What is $x$ if $\frac{1}{2}(x+6)=9$?

     $x=12$. Multiply by $2$ to get $x+6=18$, then subtract $6$.

113. What is $x$ if $x+9=2$?

     $x=-7$. Subtract $9$ from both sides: $x = 2 - 9 = -7$.

114. What property justifies multiplying both sides of $\frac{x}{5}=9$ by $5$?

     Multiplication Property of Equality. Allows multiplying both sides by the same non-zero value.

115. What is the solution to $2(x-5)+3=11$?

     $x=9$. Distribute to get $2x-10+3=11$, simplify, then solve.

116. What is the solution to $0.2x+1=3$?

     $x=10$. Subtract 1, then divide by 0.2: $2÷0.2=10$.

117. What is the solution to $5(x-2)=15$?

     $x=5$. Divide by 5, then add 2: $x - 2 = 3$, so $x = 5$.

118. What is the solution to $4(x-3)=20$?

     $x=8$. Distribute 4, add 12, divide by 4: $32÷4=8$.

119. What is the solution of $x + 7 = 20$?

     $x = 13$. Subtract 7 from both sides: $20 - 7 = 13$.

120. What is the solution set of $x^2=9$ over the real numbers?

     $x=3$ or $x=-3$. Both positive and negative square roots of $9$.

121. What is the solution to $\frac{x}{4}=9$?

     $x=36$. Multiply both sides by 4: $9×4=36$.

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

     $x=7$. Multiply by $5$ to get $2x+1=15$, then solve: $x = 7$.

123. What is the solution to $\frac{3}{x}=6$ with $x\ne 0$?

     $x=\frac{1}{2}$. Cross multiply: $3 = 6x$, so $x = \frac{3}{6} = \frac{1}{2}$.

124. What property justifies multiplying both sides of $\frac{x}{5}=9$ by $5$?

     Multiplication Property of Equality. Multiplying both sides by the same non-zero value maintains equality.

125. What property allows you to multiply both sides of $a=b$ by the same nonzero number and keep equality?

     Multiplication Property of Equality. States that multiplying both sides by the same nonzero value preserves equality.

126. What property justifies multiplying both sides of $a=b$ by the same nonzero number?

     Multiplication Property of Equality. If $a=b$ and $c≠0$, then $ac=bc$.

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

     $x=17$. Multiply by 5, then add 2: $x - 2 = 15$, so $x = 17$.

128. What is the solution of $|x - 3| = 5$?

     $x = 8$ or $x = -2$. $x - 3 = 5$ or $x - 3 = -5$, giving $x = 8$ or $x = -2$.

129. Identify the solution to $7-2x=1$.

     $x=3$. Subtract 7: $-2x=-6$, then divide by $-2$.

130. What is the solution set of $2(x+1)=2x+5$?

     No solution. Distributing gives $2x+2=2x+5$, which simplifies to $2=5$ (false).

131. What property justifies multiplying both sides of $a=b$ by the same nonzero number?

     Multiplication Property of Equality. If $a=b$ and $c≠0$, then $ac=bc$.

132. What is the solution to $\frac{2}{3}x=10$?

     $x=15$. Multiply both sides by $\frac{3}{2}$: $x = 10 cdot \frac{3}{2} = 15$.

133. What is $x$ if $4(x-2)=12$?

     $x=5$. Distribute to get $4x-8=12$, add $8$, divide by $4$.

134. Identify the solution to $3x-7=11$.

     $x=6$. Add 7 to both sides: $3x=18$, then divide by 3.

135. What is the solution to $|x-3|=7$?

     $x=10$ or $x=-4$. Absolute value gives two cases: $x-3=7$ or $x-3=-7$.

136. Identify the solution type if simplifying gives $0x=0$.

     Infinitely many solutions. Since $0$ times any $x$ equals $0$, always true.

137. What is the standard approach to solve an equation with fractions like $\frac{x}{3}+\frac{1}{2}=2$?

     Multiply both sides by the least common denominator. Eliminates fractions to simplify solving.

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

     $x=7$. Multiply by 5, then solve: $2x + 1 = 15$, so $x = 7$.

139. What is the solution to the equation $0.2x=8$?

     $x=40$. Divide by 0.2 (same as multiply by 5): $x = 8 ÷ 0.2 = 40$.

140. What is the absolute value definition for $|x|=a$ when $a>0$?

     $x=a$ or $x=-a$. The absolute value equals $a$ when $x$ is $a$ units from zero.

141. What is the solution to $3x-8=16$?

     $x=8$. Add 8, then divide by 3: $24÷3=8$.

142. Identify the solution to $\frac{1}{2}x-\frac{3}{4}=\frac{5}{4}$.

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

143. What is $x$ if $2x+5=x-1$?

     $x=-6$. Subtract $x$ and $5$ from both sides to get $x=-6$.

144. What is the first step to solve a one-variable equation efficiently?

     Simplify both sides by combining like terms. Simplifying first reduces errors and makes solving easier.

145. What is the classification of $0x=5$ as an equation in one variable?

     No solution. Since $0 cdot x = 0 eq 5$, the equation has no solution.

146. What is $x$ if $\frac{3}{4}x=12$?

     $x=16$. Multiply both sides by $\frac{4}{3}$: $x = 12 × \frac{4}{3} = 16$.

147. What is the result type if solving simplifies to $0=0$?

     Infinitely many solutions. Every value of $x$ satisfies the equation.

148. What property justifies adding the same number to both sides of $x-7=12$?

     Addition Property of Equality. Adding the same value to both sides maintains equality.

149. What is $x$ if $2x-5=11$?

     $x=8$. Add $5$ to both sides, then divide by $2$.

150. What is the solution to $7-3x=22$?

     $x=-5$. Subtract $7$ from both sides, then divide by $-3$.

151. What is the solution to $\frac{3}{4}x=18$?

     $x=24$. Multiply both sides by $\frac{4}{3}$: $18×\frac{4}{3}=24$.

152. Identify the solution to $4(x-3)=20$.

     $x=8$. Distribute: $4x-12=20$, add 12, then divide by 4.

153. What is the solution to the equation $x^2=49$?

     $x=7$ or $x=-7$. Take square root of both sides: $x = ±sqrt{49} = ±7$.

154. What is the solution to the equation $|3x|=12$?

     $x=4$ or $x=-4$. Divide by 3: $|x| = 4$, so $x = 4$ or $x = -4$.

155. What is the first step to solve $3(x-4)=21$?

     Divide both sides by $3$. Simplify by dividing to get $x-4=7$ before isolating $x$.

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

     $x=16$. Multiply by $3$ to get $x-1=15$, then add $1$: $x = 16$.

157. What is the first step to solve a linear equation with parentheses, such as $2(x-3)=10$?

     Distribute to remove parentheses. Eliminates parentheses to simplify the equation.

158. What is the solution to $\frac{3}{4}x=18$?

     $x=24$. Multiply both sides by $\frac{4}{3}$ to isolate $x$.

159. What is the solution to $3(x-2)+4=10$?

     $x=4$. Distribute, combine like terms: $3x-2=10$, so $x=4$.

160. What must you do after clearing denominators in a rational equation?

     Check solutions for extraneous values in the original equation. Some solutions may make denominators zero.

161. What is the solution to $5x=45$?

     $x=9$. Divide both sides by 5: $45÷5=9$.

162. What is the solution of $2(x+6)=20$?

     $x=4$. Distribute: $2x+12=20$, then solve for $x$.

163. What is the solution to $x+7=19$?

     $x=12$. Subtract 7 from both sides: $19-7=12$.

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

     Multiply both sides by $3$. Eliminates the denominator to simplify solving.

165. What property justifies adding the same number to both sides of an equation?

     Addition Property of Equality. Allows adding same value to both sides to maintain equality.

166. What operation preserves equivalence when solving $ax+b=c$ (for $a\ne 0$)?

     Add/subtract same value, or multiply/divide by same nonzero value. These operations maintain equality on both sides.

167. What is the solution to $|2x+1|=9$?

     $x=4$ or $x=-5$. $2x+1=9$ or $2x+1=-9$, so $x=4$ or $x=-5$.

168. What is the solution of $2(x + 3) = 14$?

     $x = 4$. Divide by 2, then subtract 3: $14 ÷ 2 - 3 = 4$.

169. What is the solution to $0.2x=6$?

     $x=30$. Divide both sides by $0.2$ (same as multiplying by $5$).

170. What is the solution to $2x-9=5$?

     $x=7$. Add $9$ to both sides, then divide by $2$.

171. Identify the property: $a(b+c)=ab+ac$.

     Distributive property. Multiplication distributes over addition.

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

     $x=7$. Multiply by 3, subtract 1, then divide by 2.

173. What is $x$ if $7-3x=19$?

     $x=-4$. Subtract $7$, then divide by $-3$.

174. What is the solution to the equation $x+7=19$?

     $x=12$. Subtract 7 from both sides: $x = 19 - 7 = 12$.

175. What property justifies multiplying both sides of $a=b$ by the same nonzero number?

     Multiplication property of equality. This property maintains equality when multiplying both sides.

176. What is the solution to $5(x+2)=3x+18$?

     $x=4$. Distribute, combine like terms: $2x=8$, so $x=4$.

177. What is $x$ if $\frac{x-2}{3}=5$?

     $x=17$. Multiply by $3$ to get $x-2=15$, then add $2$.

178. What is the solution set of $4(x - 1) = 4x - 4$?

     Infinitely many solutions. Distributing gives $4x - 4 = 4x - 4$, always true.

179. What is the solution to $\frac{x}{2}+\frac{x}{3}=10$?

     $x=12$. Common denominator 6: $\frac{3x}{6} + \frac{2x}{6} = 10$, so $\frac{5x}{6} = 10$, thus $x = 12$.

180. What is the solution to $3x-5=16$?

     $x=7$. Add 5 to both sides, then divide by 3: $21÷3=7$.

181. What is $x$ if $2(x-3)+4=10$?

     $x=6$. Distribute: $2x-6+4=10$, then $2x-2=10$, so $x=6$.

182. What is the solution of $x+7=19$?

     $x=12$. Subtract $7$ from both sides: $19-7=12$.

183. What is the solution to $|2x+1|=7$?

     $x=3$ or $x=-4$. Set $2x + 1 = 7$ or $2x + 1 = -7$, solving gives $x = 3$ or $x = -4$.

184. What is the solution to $2(x+3)-x=10$?

     $x=4$. Expand to $2x+6-x=10$, simplify to $x+6=10$, so $x = 4$.

185. What property allows you to add the same number to both sides of $a=b$ and keep equality?

     Addition Property of Equality. States that adding the same value to both sides preserves equality.

186. What is $x$ if $7-2x=3x+2$?

     $x=1$. Add $2x$ and subtract $2$ to get $5=5x$.

187. What is the solution to $3x-2=2x+6$?

     $x=8$. Subtract $2x$ from both sides: $x-2=6$, so $x = 8$.

188. What is the solution to the equation $|x-2|=5$?

     $x=7$ or $x=-3$. Absolute value gives two cases: $x - 2 = 5$ or $x - 2 = -5$.

189. What is the multiplicative property of equality used when solving $ax=b$?

     Multiply or divide both sides by the same nonzero number. This property maintains equation balance when multiplying/dividing.

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

     $x=16$. Multiply both sides by $3$, then add $1$.

191. What property justifies multiplying both sides of $\frac{x}{5}=9$ by $5$?

     Multiplication Property of Equality. Allows multiplying both sides by the same non-zero value.

192. What is the solution to $x^2-9=0$?

     $x=3$ or $x=-3$. Factor as $(x+3)(x-3)=0$, so $x=±3$.

193. Identify the solution to $0.4x+1.2=3.6$.

     $x=6$. Subtract 1.2: $0.4x=2.4$, then divide by 0.4.

194. What is the first step to solve $\frac{x}{3}+2=8$?

     Subtract $2$ from both sides. Isolate the fraction term before multiplying.

195. What is the rule for combining like terms in $ax+bx$?

     $ax+bx=(a+b)x$. Factor out the common variable $x$.

196. What is the solution to $2(x+3)=14$?

     $x=4$. Distribute, then solve: $2x+6=14$, so $2x=8$, thus $x=4$.

197. What is $x$ if $\frac{2x-1}{3}=5$?

     $x=8$. Multiply by $3$ to get $2x-1=15$, then solve.

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

     $x=13$. Multiply by 5, then add 3: $10+3=13$.

199. What is the first step to solve $3(x+4)=21$?

     Divide both sides by $3$. Simplify by dividing to get $x+4=7$ before isolating $x$.

200. What is the solution to $|2x+1|=9$?

     $x=4$ or $x=-5$. Set $2x+1=9$ or $2x+1=-9$, then solve each.

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

     $x=20$. Add 3, multiply by 4: $\frac{x}{4} = 5$, so $x = 20$.

202. What restriction must be stated for $\frac{1}{x-2}$ to be defined?

     $x\ne 2$. Division by zero is undefined.

203. What is the solution to $\frac{x-5}{3}=4$?

     $x=17$. Multiply both sides by 3, then add 5: $x-5=12$, so $x=17$.

204. What is the solution to $0.2x+1=5$?

     $x=20$. Subtract 1, divide by 0.2: $4÷0.2=20$.

205. What is the solution to $x^2-9=0$?

     $x=3$ or $x=-3$. Factor as $(x+3)(x-3)=0$, so $x=±3$.

206. What operation property justifies adding the same number to both sides of $a=b$?

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

207. What is the solution to $2(x+5)=2x+10$?

     All real numbers. Distributing gives $2x+10=2x+10$, always true.

208. What is $x$ if $3x-7=11$?

     $x=6$. Add $7$ to get $3x=18$, then divide by $3$.

209. What is the solution to $5x=35$?

     $x=7$. Divide both sides by 5: $35÷5=7$.

210. What is the solution of $5(x+1)=5x+5$?

     Infinitely many solutions. Distributing gives $5x+5=5x+5$, always true.

211. What is the solution to $\frac{x}{2}+\frac{x}{3}=10$?

     $x=12$. Common denominator 6: $\frac{3x+2x}{6}=10$, so $5x=60$.

212. What is $x$ if $3x+2=3x-5$?

     No solution. Subtracting $3x$ from both sides gives $2=-5$, which is false.

213. What is the solution to $5(x-1)=2(x+8)$?

     $x=7$. Expand: $5x-5=2x+16$, then $3x=21$, so $x = 7$.

214. What is the solution of $0.2x = 8$?

     $x = 40$. Divide both sides by 0.2: $8 ÷ 0.2 = 40$.

215. What property justifies adding the same number to both sides of $x-7=10$?

     Addition Property of Equality. Allows adding the same value to both sides to maintain equality.

216. What is the solution to $4x+3=2x+11$?

     $x=4$. Subtract $2x$ and 3 from both sides: $2x=8$, so $x=4$.

217. What is the solution to $3x=27$?

     $x=9$. Divide both sides by 3: $27÷3=9$.

218. What is the additive property of equality used when solving $x+a=b$?

     Add or subtract the same number on both sides. This property maintains equation balance when adding/subtracting.

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

     $x=-12$. Multiply both sides by $4$: $x = -3 × 4 = -12$.

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

     $x=16$. Multiply by $3$: $x-1=15$, then add $1$.

221. What is the solution to the equation $7-2x=15$?

     $x=-4$. Subtract 7, then divide by -2: $-2x = 8$, so $x = -4$.

222. What is $x$ if $4(x+1)=20$?

     $x=4$. Divide by $4$ to get $x+1=5$, then subtract $1$.

223. What is $x$ if $\frac{x-1}{2}+\frac{x+3}{2}=7$?

     $x=6$. Combine fractions: $\frac{2x+2}{2}=7$, so $2x+2=14$.

224. What is the solution to $7-2x=1$?

     $x=3$. Subtract 7 from both sides: $-2x=-6$, so $x=3$.

225. What is the solution of $4x + 6 = 2x + 18$?

     $x = 6$. Subtract $2x$ and 6 from both sides: $2x = 12$, so $x = 6$.

226. What is the solution to $x^2=49$?

     $x=7$ or $x=-7$. Take square root of both sides: $x = pmsqrt{49} = pm 7$.

227. What does it mean if solving an equation gives a false statement like $0=5$?

     No solution. The equation is never true, so no value of $x$ works.

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

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

229. What is the solution to the equation $5x=35$?

     $x=7$. Divide both sides by 5: $x = 35 ÷ 5 = 7$.

230. What is the solution to $|x-4|=7$?

     $x=11$ or $x=-3$. $x-4=7$ or $x-4=-7$, giving $x=11$ or $x=-3$.

231. What does it mean if solving an equation gives a true statement like $0=0$?

     Infinitely many solutions. The equation is always true, so any value of $x$ works.

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

     $x = 17$. Multiply by 3, then add 2: $5 × 3 + 2 = 17$.

233. What is the solution to $2x-5=13$?

     $x=9$. Add 5 to both sides, then divide by 2: $18÷2=9$.

234. What is the solution of $3(x-2)+4=2x+1$?

     $x=3$. Distribute: $3x-6+4=2x+1$, then simplify and solve.

235. What is the solution of $|2x + 1| = 7$?

     $x = 3$ or $x = -4$. $2x + 1 = 7$ or $2x + 1 = -7$, giving $x = 3$ or $x = -4$.

236. What is the solution to $4x+3=2x+15$?

     $x=6$. Subtract $2x$ and 3 from both sides: $2x=12$, so $x=6$.

237. What is the first step to solve $-3(x-4)=12$?

     Divide both sides by $-3$. Isolate the parentheses before distributing.

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