MathChapter 2: Solving Linear Equations
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Prompt
What is a solution of an equation?
💡A solution satisfies both sides.
Answer
A value that makes the original equation true when substituted for the variable.
Example\(x=4\) solves \(3x+2=14\) because \(3(4)+2=14\).
Prompt
What are equivalent equations?
💡Legal operations on both sides preserve equivalence.
Answer
Equations with the same solution set.
Example\(x+5=12\) and \(x=7\) are equivalent.
Prompt
What is the balance principle of equation solving?
💡Whatever one side receives, the other receives.
Answer
Apply the same valid operation to both complete sides to preserve equality.
ExampleFrom \(x-3=8\), add \(3\) to both sides.
Prompt
What does the Addition Property of Equality allow?
💡Add the same quantity to both sides.
Answer
If \(a=b\), then \(a+c=b+c\).
Example\(x-6=9\Rightarrow x=15\) after adding \(6\).
Prompt
What does the Subtraction Property of Equality allow?
💡Subtract the same quantity from both sides.
Answer
If \(a=b\), then \(a-c=b-c\).
Example\(y+8=21\Rightarrow y=13\) after subtracting \(8\).
Prompt
What does the Multiplication Property of Equality allow?
💡Multiply both sides by the same quantity.
Answer
If \(a=b\), then \(ac=bc\).
Example\(z/4=5\Rightarrow z=20\) after multiplying by \(4\).
Prompt
What restriction belongs to the Division Property of Equality?
💡Division by zero is undefined.
Answer
The divisor must be nonzero.
Example\(6x=30\Rightarrow x=5\) after dividing by nonzero \(6\).
Prompt
What are inverse operation pairs?
💡Each pair undoes the other.
Answer
Addition/subtraction and multiplication/division.
ExampleUndo \(+7\) with \(-7\), and undo \(\times5\) with \(\div5\).
Prompt
What is a reliable order for solving a multi-step linear equation?
💡Structure first, equality operations second.
Answer
Simplify each side, clear awkward fractions if useful, isolate the variable, then verify.
ExampleDistribute and combine before solving \(3(x+2)+2x=31\).
Prompt
When can dividing before distributing be efficient?
💡Choose the shorter legal path.
Answer
When both sides share the outside factor and it is nonzero.
ExampleIn \(4(x-2)=28\), divide by \(4\) first to get \(x-2=7\).
Prompt
What is the key distribution sign rule?
💡Do not lose a negative sign.
Answer
Multiply every term inside grouping by the complete signed outside factor.
Example\(-2(3x-5)=-6x+10\).
Prompt
How can denominators be cleared from a linear equation?
💡The factor must reach all terms.
Answer
Multiply every term on both sides by a common denominator.
ExampleMultiplying \(x/3+2=7\) by \(3\) gives \(x+6=21\).
Prompt
How is a fractional coefficient undone?
💡A number times its reciprocal is \(1\).
Answer
Multiply both sides by its reciprocal, provided the coefficient is nonzero.
Example\(\frac34x=9\Rightarrow x=12\) after multiplying by \(\frac43\).
Prompt
How can decimal equations be made easier to solve?
💡Move the decimal consistently on both sides.
Answer
Multiply every term by a suitable power of ten to create integer coefficients.
Example\(0.4x+0.7=2.3\) becomes \(4x+7=23\) after multiplying by \(10\).
Prompt
Why is ‘move a term and change its sign’ a risky shortcut?
💡Write the same addition or subtraction on both sides.
Answer
It hides the equality operation and makes sign errors harder to detect.
ExampleFrom \(3x+8=20\), subtract \(8\) from each side.
Prompt
When is a linear equation fully solved for \(x\)?
💡Do not stop at \(ax=b\).
Answer
When it is written \(x=k\), with \(x\) isolated and coefficient \(1\).
Example\(4x=18\Rightarrow x=9/2\).
Prompt
How should a solution be checked?
💡Use the unsimplified original to audit every step.
Answer
Substitute it into the original equation and verify that both sides have equal value.
ExampleFor \(x=-2\) in \(5x+3=-7\), both sides equal \(-7\).
Prompt
What error occurs when only one term is multiplied while clearing a denominator?
💡Distribute the clearing factor everywhere.
Answer
Equality is not preserved because the operation was not applied to the entire side.
Example\(3(x/3+2)=3(7)\) gives \(x+6=21\), not \(x+2=21\).
Prompt
Why is exact arithmetic preferable during equation solving?
💡Keep fractions as fractions when practical.
Answer
It avoids rounding error and preserves an exact solution until an approximation is requested.
ExampleReport \(x=7/3\), not an early rounded \(2.33\), unless decimals are requested.
Prompt
What should you do after solving when the problem asks for a different expression?
💡Do not automatically report the variable value.
Answer
Substitute the solution into the requested expression and evaluate it.
ExampleIf \(x=4\) and the question asks for \(3x-1\), report \(11\).
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