MathChapter 2: Solving Linear Equations
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Prompt
What is the goal when a variable appears on both sides?
💡Create an equivalent equation with no variable term on one side.
Answer
Use equality properties to collect all variable terms on one side, then isolate the variable.
Example\(7x+2=3x+18\Rightarrow4x+2=18\) after subtracting \(3x\).
Prompt
Can variable terms be collected on either side of an equation?
💡The cleaner side is a strategy choice, not a rule.
Answer
Yes. Either choice is valid if the same legal operation is applied to both sides.
ExampleBoth paths solve \(5x+1=2x+13\) as \(x=4\).
Prompt
What is the recommended six-step procedure?
💡Simplify structure before isolation.
Answer
Distribute, combine each side, collect variables, collect constants, solve, and verify.
ExampleUse the process on \(3(x+2)=2x+11\).
Prompt
Why often collect variables on the side with the larger coefficient?
💡This is convenient, not mandatory.
Answer
It commonly leaves a positive coefficient and reduces sign errors.
ExampleIn \(9x-2=4x+18\), subtract \(4x\) to leave \(5x\).
Prompt
What really happens when a term seems to ‘cross’ the equal sign?
💡Terms do not jump automatically.
Answer
Its opposite is added to both sides, preserving equality.
ExampleSubtract \(3x\) from both sides of \(8x=3x+20\).
Prompt
May like terms be combined across the equal sign?
💡The equal sign separates two expressions.
Answer
No. First use an equality operation to place them on the same side.
ExampleFrom \(6x=2x+12\), subtract \(2x\) before combining to get \(4x=12\).
Prompt
What should happen before variable terms are collected if parentheses are present?
💡Simplify left and right independently.
Answer
Distribute fully and combine like terms on each side.
Example\(2(x+3)=x+10\Rightarrow2x+6=x+10\).
Prompt
How should a negative outside factor be distributed?
💡Track both magnitude and sign.
Answer
Multiply every inside term by the complete negative factor.
Example\(-3(2x-5)=-6x+15\).
Prompt
How are fractions handled when variables appear on both sides?
💡Use one consistent legal method.
Answer
Collect fractional coefficients directly or multiply every term by a common denominator.
ExampleMultiplying \(x/3=x/6+4\) by \(6\) gives \(2x=x+24\).
Prompt
How can decimals be cleared from both sides?
💡Do not scale only selected terms.
Answer
Multiply every term on both sides by a suitable power of ten.
Example\(0.7x-1.8=0.2x+4.7\) becomes \(7x-18=2x+47\).
Prompt
What is a safe way to collect a negative variable term?
💡This can leave a positive coefficient.
Answer
Add its opposite to both sides.
ExampleFor \(9-2x=x\), add \(2x\) to get \(9=3x\).
Prompt
How do you verify a variables-on-both-sides solution?
💡Check before any distribution or collection.
Answer
Substitute it into the original equation and compare the two complete sides.
ExampleFor \(x=5\) in \(3x+4=2x+9\), both sides equal \(19\).
Prompt
What is the first algebraic priority in a nested equation?
💡Do not collect variable terms hidden inside grouping.
Answer
Simplify from the innermost grouping outward on each side.
ExampleSimplify \(2[3(x+1)-4]\) before comparing coefficients.
Prompt
How can you recognize an illegal sign change?
💡Every sign change needs an algebraic reason.
Answer
No matching addition or subtraction was applied to both sides.
ExampleFrom \(5x+2=2x+20\), subtract \(2x\) visibly.
Prompt
How is a break-even problem modeled?
💡Equal outputs create an equation with variables on both sides.
Answer
Set the two complete cost expressions equal and solve for the shared input.
Example\(18+6m=42+3m\) compares two monthly plans.
Prompt
What does ‘five less than four times \(n\)’ become?
💡Build four times the number, then subtract five.
Answer
\(4n-5\)
ExampleIt can form one side of \(2n+7=4n-5\).
Prompt
Why keep grouped numerators intact when clearing fractions?
💡Grouping prevents partial multiplication.
Answer
The clearing factor must distribute to every term in each numerator.
Example\(6[(x+1)/2]=3(x+1)\).
Prompt
If collection produces a negative coefficient, is the method wrong?
💡A positive coefficient is convenient, not required.
Answer
No. Continue legally and divide by the negative coefficient; another path gives the same solution.
Example\(-4x=-20\Rightarrow x=5\).
Prompt
What should you do when a known parameter appears with \(x\)?
💡Known letters act like known numbers.
Answer
Substitute the parameter's given value, simplify, and then collect the \(x\)-terms.
ExampleIf \(a=3\), then \(a(2x-5)=4x+9\) becomes \(3(2x-5)=4x+9\).
Prompt
What final audit catches both translation and algebra errors?
💡A numerical match and sensible units are both useful.
Answer
Check the solution in the original equation and interpret it in the original context.
ExampleA negative month count would signal a context problem even if arithmetic was legal.
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