Algebra Flashcards: Explaining And Justifying Equation Solving Steps

What this deck covers

This deck focuses on Explaining And Justifying Equation Solving Steps, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

Flashcard 3: What is the first recommended step to clear denominators in rac{x}{3}+2=5?

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

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

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

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

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

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

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

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

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

Flashcard 8: What restriction is required to use the Division Property of Equality on rac{a}{c}=rac{b}{c}?

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

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

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

Flashcard 10: What restriction is required before multiplying an equation by rac{1}{x}?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: Distributive Property. This property allows factoring out common terms.

Flashcard 19: Identify the solution of rac{x}{6}=-3.

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

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

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

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

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

Flashcard 22: What property justifies subtracting the same number from both sides of $a=b$ to get $a-c=b-c$?

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

Flashcard 23: Identify the solution of $7-2x=1$.

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

Flashcard 24: Find and correct the invalid step: from $x=3$ conclude rac{1}{x}=rac{1}{3} without a condition.

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

Flashcard 25: Which property justifies rewriting $x+0$ as $x$ during equation solving?

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

Flashcard 26: Find and correct the invalid step: from $x=3$ conclude $\frac{1}{x}=\frac{1}{3}$ without a condition.

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

Flashcard 27: What restriction is required to use the Division Property of Equality on rac{a}{c}=rac{b}{c}?

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

Flashcard 28: What property states that if $a=b$, then $a$ may be replaced by $b$ in any expression?

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

Flashcard 29: Identify the solution of rac{x}{4}+3=5.

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

Flashcard 30: Which property justifies rewriting $x+(-x)$ as $0$ during equation solving?

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

Flashcard 31: What property justifies adding the same number to both sides of $a=b$ to get $a+c=b+c$?

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

Flashcard 32: Which property justifies going from $2x+6=10$ to $2x=4$?

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

Flashcard 33: What property states that if $a=b$ and $b=c$, then $a=c$?

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

Flashcard 34: Identify the solution of $3x-2=3x+4$.

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

Flashcard 35: Identify the solution of $9=3x+0$.

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

Flashcard 36: Identify the property used to justify from $x-4=9$ to $x-4+4=9+4$.

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

Flashcard 37: What property justifies subtracting the same number from both sides of $a=b$ to get $a-c=b-c$?

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

Flashcard 38: Identify the solution of $0.2x+1=3$.

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

Flashcard 39: Identify the solution of $-3x=12$.

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

Flashcard 40: What is the standard goal when solving a linear equation in one variable?

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

Flashcard 41: Which step correctly clears denominators in $\frac{x}{2}+\frac{x}{3}=5$?

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

Flashcard 42: Identify the property used to justify from $x+2=7$ to $x=5$.

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

Flashcard 43: Identify the solution of $9=3x+0$.

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

Flashcard 44: Which operation is typically used first to solve $x-\frac{3}{2}=4$?

Answer: Add rac{3}{2} to both sides. Adding the opposite of the subtracted term isolates $x$.

Flashcard 45: Identify the solution of $5x-1=2x+8$.

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

Flashcard 46: Identify the solution of $4x=28$.

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

Flashcard 47: Identify the solution of rac{x}{2}+rac{x}{3}=5.

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

Flashcard 48: Identify the solution of rac{2x-1}{3}=5.

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

Flashcard 49: Identify the solution of $0.5x=6$.

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

Flashcard 50: Identify the solution of $2(x+4)=18$.

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

Flashcard 51: What inverse operation undoes adding $k$ in $x+k$?

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

Flashcard 52: Which step correctly clears denominators in rac{x}{2}+rac{x}{3}=5?

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

Flashcard 53: What conclusion is justified if solving leads to $0=7$?

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

Flashcard 54: Which property justifies going from $2x=4$ to $x=2$?

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

Flashcard 55: What property states that if $a=b$, then $b=a$?

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

Flashcard 56: Identify the solution of $3x+5=20$.

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

Flashcard 57: What property states that if $a=b$ and $b=c$, then $a=c$?

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

Flashcard 58: Identify the solution of $0.5x=6$.

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

Flashcard 59: What property justifies multiplying both sides of $a=b$ by $c$ to get $ac=bc$?

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

Flashcard 60: Identify the solution set of $2(x+1)=2x+2$.

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

Flashcard 61: What is the most direct way to check that $x=4$ solves $2x+1=9$?

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

Flashcard 62: Identify the solution of $4x=28$.

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

Flashcard 63: Identify the solution of $3(x-1)+2=11$.

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

Flashcard 64: What term describes an operation that reverses another operation when solving equations?

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

Flashcard 65: What is the most direct way to check that $x=4$ solves $2x+1=9$?

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

Flashcard 66: Identify the solution of $2x+6=x+10$.

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

Flashcard 67: Find and correct the invalid step: divide both sides of $x\cdot 0=5\cdot 0$ by $0$ to get $x=5$.

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

Flashcard 68: Identify the property used to justify from $3(x+2)=21$ to $x+2=7$.

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