MathChapter 3: Functions and Linear Equations
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Prompt
What is a system of linear equations?
💡A solution satisfies all equations.
Answer
A set of linear equations involving the same variables and solved simultaneously.
ExampleTwo equations in x and y form a common SAT system.
Prompt
What does a system solution represent graphically?
💡Usually an intersection.
Answer
A point shared by every graph in the system.
ExampleSubstitute it into both equations.
Prompt
When does a two-line system have one solution?
💡One shared point.
Answer
When the lines have different slopes and intersect once.
ExampleThe ordered pair satisfies both equations.
Prompt
When does a system have no solution?
💡They are parallel.
Answer
When distinct lines have equal slopes and different intercepts.
ExampleElimination may produce a false statement.
Prompt
When does a system have infinitely many solutions?
💡Slope and intercept both match.
Answer
When both equations describe the same line.
ExampleElimination may produce \(0=0\).
Prompt
List the substitution method steps.
💡Use parentheses around substituted expressions.
Answer
Isolate one variable, substitute, solve, back-substitute, and verify.
ExampleBest when a variable is already isolated.
Prompt
List the elimination method steps.
💡Multiply every term when scaling.
Answer
Align terms, scale if needed, add/subtract to eliminate, solve, back-substitute, verify.
ExampleCreate opposite coefficients.
Prompt
When is graphing most useful for systems?
💡It may be approximate.
Answer
For visualizing solution count or reading a clean exact intersection.
ExampleUse algebra for exact fractional intersections.
Prompt
What does a false statement after elimination mean?
💡The variables canceled from parallel lines.
Answer
No solution.
Example\(0=7\) cannot be repaired by choosing x or y.
Prompt
What does a true statement after elimination mean?
Answer
The equations are equivalent and have infinitely many solutions.
Example\(0=0\) confirms dependence.
Prompt
Why back-substitute after elimination?
💡A single number is incomplete.
Answer
Elimination finds one coordinate; the other is needed for the ordered-pair solution.
ExampleThen verify both equations.
Prompt
What is the main scaling trap in elimination?
💡Distribute the multiplier.
Answer
Multiplying only one term instead of every term in an equation.
Example\(2(x+3y)=2x+6y\).
Prompt
What is the main substitution sign trap?
💡Replace the whole variable value.
Answer
Failing to parenthesize a multi-term or negative substituted expression.
Example\(-2)(3x-5)\) needs distribution.
Prompt
How do you choose a parameter for no solution?
💡Rewrite both lines comparably.
Answer
Make slopes equal while keeping intercepts different.
ExampleMatching only x-coefficients may be misleading.
Prompt
How do you choose parameters for infinitely many solutions?
💡Equations must be scalar multiples.
Answer
Make the simplified slope and intercept both match.
ExampleEvery term must share the same scale factor.
Prompt
How do you verify an ordered-pair system solution?
💡Both must be true.
Answer
Substitute x and y into every original equation.
ExampleSatisfying only one equation is insufficient.
Prompt
When is substitution usually efficient?
💡Avoid unnecessary fractions.
Answer
When one variable is isolated or has coefficient 1 or -1.
ExampleChoose the simpler equation.
Prompt
When is elimination usually efficient?
💡Align like terms.
Answer
When coefficients are already opposite or can become opposite with small multipliers.
ExampleStandard form helps.
Prompt
What does consistent and independent mean for a two-line system?
💡The lines intersect.
Answer
Exactly one solution.
ExampleDifferent slopes guarantee this.
Prompt
What does dependent mean for a linear system?
💡One equation adds no new restriction.
Answer
The equations describe the same line, so there are infinitely many solutions.
ExampleEquivalent equations are dependent.
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