MathAlgebra
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Card 1 of 12. Answer hidden.
Prompt
What is the goal when solving a linear equation in one variable?
💡Think about what must remain balanced.
Answer
Isolate the variable while keeping both sides of the equation equivalent.
Example\(3x+5=20\;\Rightarrow\;x=5\)
Prompt
Which operations undo addition and multiplication?
Answer
Inverse operations: subtraction undoes addition, and division undoes multiplication.
Prompt
What must happen when a factor is distributed across parentheses?
💡Do not stop after the first term.
Answer
Multiply the factor by every term inside the parentheses.
Prompt
What does it mean if simplifying an equation produces a true statement such as \(0=0\)?
Answer
The equation is an identity and has infinitely many solutions.
Prompt
What does it mean if the variable terms cancel and leave a false statement such as \(0=7\)?
💡A false numerical statement can never be made true.
Answer
The equation is inconsistent and has no solution.
Prompt
How can a fixed fee plus a constant rate be modeled?
Answer
Use a linear equation: total equals the fixed amount plus rate times quantity.
ExampleA $5 fee plus $2.40 per mile becomes \(T=5+2.40m\).
Prompt
What is a reliable way to check a proposed solution?
Answer
Substitute the value into the original equation and verify that both sides are equal.
ExampleFor \(x=6\) in \(4x+7=31\), both sides equal \(31\).
Prompt
What SAT trap can occur when variables appear on both sides?
💡An equation must stay balanced at every step.
Answer
Moving a term without applying the same operation to both sides can change the equation.
ExampleFrom \(5x-9=2x+18\), subtract \(2x\) from both sides before continuing.
Prompt
What is the fastest first step for an equation containing several fractions?
💡Remove the denominators without changing the solution.
Answer
Multiply every term on both sides by the least common denominator.
Prompt
What should you check before solving for \(x\) when a question asks for an expression involving \(x\)?
Answer
Check whether the requested expression is a direct multiple or rearrangement of the given expression.
ExampleIf \(7x+4=39\), then \(14x+8=2(39)=78\).
Prompt
In a contextual equation such as \(8s+12a=240\), what do the coefficients usually represent?
Answer
Rates or prices per unit. Here, $8 is the price per student ticket and $12 is the price per adult ticket.
Prompt
How do coefficients determine whether \(kx+b=mx+c\) has no solution or infinitely many solutions?
💡Compare variable coefficients first, then constants.
Answer
If \(k=m\) and \(b=c\), there are infinitely many solutions. If \(k=m\) but \(b\ne c\), there is no solution.
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