Systems and Linear Inequalities flashcards

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Substitute into a target expressionSolving Systems by Substitution: Solve the pair, then answer the requested combination.
When should I use direct substitution?Solving Systems by Substitution: Substitute an isolated variable into another equation.
What method supports substitute into a target expression?Solving Systems by Substitution: If y = kx, then ax + by = (a + bk)x
Isolate before substitutingSolving Systems by Substitution: Choose the easiest variable to isolate.
Direct substitutionSolving Systems by Substitution: Replace the variable with what it equals.
What must I check in solving systems by substitution?Solving Systems by Substitution: A solution is an ordered pair satisfying both original equations
What method supports isolate before substituting?Solving Systems by Substitution: x + y = s ⇒ y = s − x
What method supports direct substitution?Solving Systems by Substitution: y = mx + b and ax + y = c ⇒ (a + m)x + b = c
When should I use substitute into a target expression?Solving Systems by Substitution: Find a combination of the solution coordinates.
When should I use isolate before substituting?Solving Systems by Substitution: Rewrite one equation to make substitution efficient.
Eliminate to find a combinationSolving Systems by Elimination: Look for the target hidden in a sum of equations.
When should I use immediate elimination?Solving Systems by Elimination: Add equations whose variable coefficients are opposites.
What method supports eliminate to find a combination?Solving Systems by Elimination: If kx + ky = c, then x + y = c/k, k ≠ 0
Scale and eliminateSolving Systems by Elimination: Make matching coefficients before canceling.
Immediate eliminationSolving Systems by Elimination: Add the equations so one variable disappears.
What must I check in solving systems by elimination?Solving Systems by Elimination: Scale every term of an equation before adding; retain signs when subtracting
What method supports scale and eliminate?Solving Systems by Elimination: Multiply an entire equation by a nonzero constant
What method supports immediate elimination?Solving Systems by Elimination: (ax + by) + (cx − by) = (a + c)x
When should I use eliminate to find a combination?Solving Systems by Elimination: Use equation sums to obtain a requested expression directly.
When should I use scale and eliminate?Solving Systems by Elimination: Multiply an equation to create canceling coefficients.
Parameter for a coincident lineGraphical Solutions and Number of Solutions: Every term must scale by the same factor.
When should I use unique intersection?Graphical Solutions and Number of Solutions: Identify when two linear graphs meet once.
What method supports parameter for a coincident line?Graphical Solutions and Number of Solutions: Ax + By = C and kAx + kBy = kC describe the same line, k ≠ 0
Parallel versus identical graphsGraphical Solutions and Number of Solutions: Same direction plus a different starting value means no crossing.
Unique intersectionGraphical Solutions and Number of Solutions: Different slopes lead to one crossing.
What must I check in graphical solutions and number of solutions?Graphical Solutions and Number of Solutions: An intersection must lie on both lines; matching slopes alone do not distinguish parallel from identical lines
What method supports parallel versus identical graphs?Graphical Solutions and Number of Solutions: same m, different b ⇒ no solution; same m and b ⇒ infinitely many
What method supports unique intersection?Graphical Solutions and Number of Solutions: m₁ ≠ m₂ ⇒ exactly one intersection
When should I use parameter for a coincident line?Graphical Solutions and Number of Solutions: Match all coefficients to obtain infinitely many system solutions.
When should I use parallel versus identical graphs?Graphical Solutions and Number of Solutions: Classify systems by comparing normalized equations.
Two rates observed twiceSystems of Equations Word Problems: Different combinations reveal the individual rates.
When should I use count and value systems?Systems of Equations Word Problems: Model a total count and a total value.
What method supports two rates observed twice?Systems of Equations Word Problems: ax + by = c and dx + ey = f
Mixture systemsSystems of Equations Word Problems: Track total mixture and total ingredient separately.
Count and value systemsSystems of Equations Word Problems: Use one equation for how many and another for how much.
What must I check in systems of equations word problems?Systems of Equations Word Problems: Unknown counts should be nonnegative integers and must satisfy both contextual totals
What method supports mixture systems?Systems of Equations Word Problems: x + y = T; p₁x + p₂y = pT
What method supports count and value systems?Systems of Equations Word Problems: x + y = N; ax + by = V
When should I use two rates observed twice?Systems of Equations Word Problems: Recover rates from two different resource combinations.
When should I use mixture systems?Systems of Equations Word Problems: Use weighted contributions to model a blend.
Integer constraintsLinear Inequalities in One Variable: Solve first, then enforce whole-item feasibility.
When should I use positive coefficient inequalities?Linear Inequalities in One Variable: Solve and interpret a one-variable inequality.
What method supports integer constraints?Linear Inequalities in One Variable: x ≤ B/p with integer x ⇒ maximum x = floor(B/p)
Negative coefficient inequalitiesLinear Inequalities in One Variable: Dividing by a negative flips the arrow.
Positive coefficient inequalitiesLinear Inequalities in One Variable: Balance both sides and keep the arrow when dividing by a positive number.
What must I check in linear inequalities in one variable?Linear Inequalities in One Variable: Reverse an inequality when multiplying or dividing by a negative number
What method supports negative coefficient inequalities?Linear Inequalities in One Variable: −ax < b ⇒ x > −b/a for a > 0
What method supports positive coefficient inequalities?Linear Inequalities in One Variable: ax + b ≤ c ⇒ x ≤ (c − b)/a for a > 0
When should I use integer constraints?Linear Inequalities in One Variable: Use an inequality to find a feasible integer maximum.
When should I use negative coefficient inequalities?Linear Inequalities in One Variable: Reverse direction after division by a negative.
Maximum feasible integer valueLinear Inequalities in Two Variables and Systems of Inequalities: Apply all the limits, then choose the largest allowed integer.
When should I use test an ordered pair?Linear Inequalities in Two Variables and Systems of Inequalities: Determine whether a point satisfies a linear inequality.
What method supports maximum feasible integer value?Linear Inequalities in Two Variables and Systems of Inequalities: ax + by ≤ C ⇒ y ≤ (C − ax)/b when b > 0
Intersect two half-planesLinear Inequalities in Two Variables and Systems of Inequalities: A system uses AND: every inequality must be true.
Test an ordered pairLinear Inequalities in Two Variables and Systems of Inequalities: Plug in the point and test the statement.
What must I check in linear inequalities in two variables and systems of inequalities?Linear Inequalities in Two Variables and Systems of Inequalities: A boundary line is included for ≤ or ≥ and excluded for < or >; a system requires all conditions
What method supports intersect two half-planes?Linear Inequalities in Two Variables and Systems of Inequalities: solution region = intersection of individual solution sets
What method supports test an ordered pair?Linear Inequalities in Two Variables and Systems of Inequalities: (a,b) satisfies y > mx + c iff b > ma + c
When should I use maximum feasible integer value?Linear Inequalities in Two Variables and Systems of Inequalities: Combine linear inequalities and discrete restrictions.
When should I use intersect two half-planes?Linear Inequalities in Two Variables and Systems of Inequalities: Require a point to satisfy every inequality in a system.