Linear Equations in One Variable flashcards

0/60

60 cards remaining

Take the chapter practice test

All 60 cards in this deckShow

Why check the original equation?Solving Multistep Linear Equations: It detects sign, arithmetic, or transcription errors; a valid solution makes both original sides equal.
Distribute −3 across 2x − 5.Solving Multistep Linear Equations: −6x + 15. Multiply −3 by each term, including −5.
When is algebra especially efficient?Solving Multistep Linear Equations: When a short linear equation can be simplified and solved exactly with a few operations.
What is a solution of an equation?Solving Multistep Linear Equations: A value that makes the original left and right sides equal.
What must you do when dividing a whole equation?Solving Multistep Linear Equations: Divide every term on both sides by the same nonzero number.
Desmos intersection: which coordinate solves L(x) = R(x)?Solving Multistep Linear Equations: The x-coordinate. The y-coordinate is their common output.
Can 3x and 5 be combined into 8x?Solving Multistep Linear Equations: No. A variable term and a constant are unlike terms.
How do you solve ax + b = c when a ≠ 0?Solving Multistep Linear Equations: Subtract b from both sides, then divide both sides by a: x = (c − b)/a.
If the question asks for 3x + 2, when are you finished?Solving Multistep Linear Equations: After substituting the solved x into 3x + 2 and evaluating it.
Why does adding the same number to both sides work?Solving Multistep Linear Equations: Equal quantities remain equal after the same addition; the equation keeps the same solutions.
Decimal coefficientsEquations with Fractions and Decimals: Shift every term the same number of decimal places.
When should I use clear one denominator?Equations with Fractions and Decimals: Solve a linear equation containing a fraction.
What method supports decimal coefficients?Equations with Fractions and Decimals: 0.1ax + 0.1b = 0.1c ⇔ ax + b = c
Clear multiple fractionsEquations with Fractions and Decimals: One common multiplier can remove all numerical denominators.
Clear one denominatorEquations with Fractions and Decimals: Multiply the whole equation by the denominator.
What must I check in equations with fractions and decimals?Equations with Fractions and Decimals: Every denominator is nonzero; clearing fractions multiplies every term, not just fractions
What method supports clear multiple fractions?Equations with Fractions and Decimals: x/a + b/c = d ⇒ cx + ab = acd
What method supports clear one denominator?Equations with Fractions and Decimals: (x + a)/b = c ⇒ x + a = bc
When should I use decimal coefficients?Equations with Fractions and Decimals: Convert decimal coefficients to integers before solving.
When should I use clear multiple fractions?Equations with Fractions and Decimals: Use a least common denominator on both sides.
Negative distributionVariables on Both Sides and Distributive Equations: A minus outside parentheses changes each sign inside.
When should I use collect variables?Variables on Both Sides and Distributive Equations: Solve a linear equation with variables on both sides.
What method supports negative distribution?Variables on Both Sides and Distributive Equations: −a(bx − c) = −abx + ac
Distribute then collectVariables on Both Sides and Distributive Equations: Open the group, then collect matching pieces.
Collect variablesVariables on Both Sides and Distributive Equations: Bring variable terms together with one operation on both sides.
What must I check in variables on both sides and distributive equations?Variables on Both Sides and Distributive Equations: Combine only like terms; distribute negative factors to every term
What method supports distribute then collect?Variables on Both Sides and Distributive Equations: a(x + b) = ax + ab
What method supports collect variables?Variables on Both Sides and Distributive Equations: ax + b = cx + d ⇒ (a − c)x = d − b
When should I use negative distribution?Variables on Both Sides and Distributive Equations: Handle a subtracted expression on both sides of an equation.
When should I use distribute then collect?Variables on Both Sides and Distributive Equations: Solve an equation containing a grouped expression and variables on both sides.
Parameter conditionsOne Solution, No Solution, and Infinitely Many Solutions: For every x to work, match both the x-part and the number-part.
When should I use unique solutions?One Solution, No Solution, and Infinitely Many Solutions: Recognize a nonzero remaining coefficient.
What method supports parameter conditions?One Solution, No Solution, and Infinitely Many Solutions: a(x + b) = ax + k for all x ⇔ k = ab
Contradictions and identitiesOne Solution, No Solution, and Infinitely Many Solutions: When x cancels, ask whether the leftover statement is true.
Unique solutionsOne Solution, No Solution, and Infinitely Many Solutions: A nonzero multiple of x equals a number, so x has one value.
What must I check in one solution, no solution, and infinitely many solutions?One Solution, No Solution, and Infinitely Many Solutions: Classify after simplification; never divide by a coefficient that becomes zero
What method supports contradictions and identities?One Solution, No Solution, and Infinitely Many Solutions: 0x = k: no solutions if k ≠ 0; all real x if k = 0
What method supports unique solutions?One Solution, No Solution, and Infinitely Many Solutions: a ≠ c ⇒ one solution
When should I use parameter conditions?One Solution, No Solution, and Infinitely Many Solutions: Choose a constant that makes an equation an identity.
When should I use contradictions and identities?One Solution, No Solution, and Infinitely Many Solutions: Distinguish no solutions from infinitely many.
Unknown in several termsRearranging Formulas and Literal Equations: Gather every copy of the target letter, then factor it out.
When should I use isolate a numerator variable?Rearranging Formulas and Literal Equations: Rearrange a formula by inverse operations.
What method supports unknown in several terms?Rearranging Formulas and Literal Equations: px + qx = r ⇒ x = r/(p + q), p + q ≠ 0
Formulas with productsRearranging Formulas and Literal Equations: Divide by all of the factors attached to the target.
Isolate a numerator variableRearranging Formulas and Literal Equations: Choose one letter as the target and undo its surrounding operations.
What must I check in rearranging formulas and literal equations?Rearranging Formulas and Literal Equations: Record nonzero denominators and coefficients before division
What method supports formulas with products?Rearranging Formulas and Literal Equations: A = bh/2 ⇒ h = 2A/b, b ≠ 0
What method supports isolate a numerator variable?Rearranging Formulas and Literal Equations: y = mx + b ⇒ x = (y − b)/m, m ≠ 0
When should I use unknown in several terms?Rearranging Formulas and Literal Equations: Factor the requested variable before dividing.
When should I use formulas with products?Rearranging Formulas and Literal Equations: Solve a product formula for a selected factor.
Related quantities and a constraintLinear Equation Word Problems: Use the relationship to avoid introducing an unnecessary second unknown.
When should I use starting amount and constant change?Linear Equation Word Problems: Model a fixed starting amount and repeated equal additions.
What method supports related quantities and a constraint?Linear Equation Word Problems: If B = A + d and A + B = T, then 2A + d = T
Compare two plansLinear Equation Word Problems: Write both totals using one variable, then make them equal.
Starting amount and constant changeLinear Equation Word Problems: Separate what you start with from what changes each time.
What must I check in linear equation word problems?Linear Equation Word Problems: Define the unknown and check nonnegative or integer constraints when the context requires them
What method supports compare two plans?Linear Equation Word Problems: a + bx = c + dx
What method supports starting amount and constant change?Linear Equation Word Problems: final = initial + rate × intervals
When should I use related quantities and a constraint?Linear Equation Word Problems: Express related unknown quantities with one variable.
When should I use compare two plans?Linear Equation Word Problems: Set equal two expressions that describe the same total.