Card 1 of 20
EasyStudy card 1
Prompt
What is a literal equation?
Many formulas are literal equations.
Answer
An equation containing two or more variables that expresses a general relationship.
Example
\(d=rt\) relates distance, rate, and time.
MathChapter 2: Solving Linear Equations
Flashcards
Solving for a Specific Variable Flashcards
Twenty concise recall cards for solving for a specific variable, including rules, recognition, examples, traps, and SAT strategy.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What is the target-variable mindset?
Solve only for the letter named.
Answer
Circle the requested variable and treat every other symbol as a constant while isolating it.
Example
In \(Ax+B=C\) for \(x\), \(A,B,C\) act like fixed numbers.
Card 3 of 20
EasyStudy card 3
Prompt
Solve \(Ax+B=C\) for \(x\).
Subtract \(B\), then divide the complete side by \(A\).
Answer
\(x=(C-B)/A\), assuming \(A\ne0\).
Example
Do not write \(C-B/A\), which divides only \(B\).
Card 4 of 20
EasyStudy card 4
Prompt
Solve \(y=mx+b\) for \(x\).
Undo addition before multiplication.
Answer
\(x=(y-b)/m\), assuming \(m\ne0\).
Example
First write \(y-b=mx\).
Card 5 of 20
EasyStudy card 5
Prompt
Solve \(d=rt\) for \(t\).
Divide by the target's coefficient.
Answer
\(t=d/r\), assuming \(r\ne0\).
Example
Distance divided by rate has units of time.
Card 6 of 20
MediumStudy card 6
Prompt
How should a whole multi-term side be divided?
A fraction bar acts as grouping.
Answer
Keep the complete side grouped in one numerator or distribute the division to every term correctly.
Example
\((P-2L)/2=P/2-L\).
Card 7 of 20
MediumStudy card 7
Prompt
What should happen when the target appears in multiple terms?
Use the distributive property in reverse.
Answer
Collect all target terms on one side, then factor the target before dividing.
Example
\(xr+xs=x(r+s)\).
Card 8 of 20
MediumStudy card 8
Prompt
Solve \(P=xr+xs\) for \(x\).
Factor \(x\) first.
Answer
\(x=P/(r+s)\), assuming \(r+s\ne0\).
Example
\(P=x(r+s)\) is the key intermediate form.
Card 9 of 20
MediumStudy card 9
Prompt
Solve \(ax+b=cx+d\) for \(x\).
Collect target terms and constants before factoring.
Answer
\(x=(d-b)/(a-c)\), provided \(a\ne c\).
Example
\(x(a-c)=d-b\).
Card 10 of 20
HardStudy card 10
Prompt
Why does \(a\ne c\) matter in \(x=(d-b)/(a-c)\)?
Never divide by zero.
Answer
The rearrangement divides by \(a-c\); if it is zero, the original equation must be classified as identity or no solution.
Example
When \(a=c\), compare \(b\) and \(d\) instead.
Card 11 of 20
MediumStudy card 11
Prompt
How do you isolate a target in a denominator?
Record original nonzero restrictions.
Answer
First multiply to clear the target-containing denominator, then isolate the resulting target product.
Example
\(R=D/t\Rightarrow Rt=D\Rightarrow t=D/R\).
Card 12 of 20
MediumStudy card 12
Prompt
Solve \(C=\frac59(F-32)\) for \(F\).
Undo the outside fraction before adding \(32\).
Answer
\(F=\frac95C+32\).
Example
Multiply by \(9/5\), then add \(32\).
Card 13 of 20
MediumStudy card 13
Prompt
Solve \(A=bh/2\) for \(h\).
Clear the denominator two first.
Answer
\(h=2A/b\), assuming \(b\ne0\).
Example
\(2A=bh\Rightarrow h=2A/b\).
Card 14 of 20
MediumStudy card 14
Prompt
Solve \(P=2L+2W\) for \(W\).
Subtract the non-target term, then divide the complete side.
Answer
\(W=(P-2L)/2=P/2-L\).
Example
Both displayed forms are equivalent.
Card 15 of 20
MediumStudy card 15
Prompt
Why is a symbolic expression a complete answer?
Do not invent values.
Answer
Without numerical inputs, the task is to express the target's relationship to the other variables, not calculate a decimal.
Example
Solving \(A=LW\) for \(W\) ends at \(W=A/L\).
Card 16 of 20
HardStudy card 16
Prompt
What restriction should be checked after a literal rearrangement?
Inspect the final fraction and original domain.
Answer
Any expression used as a denominator must be nonzero, along with restrictions from the original equation.
Example
\(x=P/(r+s)\) requires \(r+s\ne0\).
Card 17 of 20
HardStudy card 17
Prompt
How do you solve a reciprocal formula for one denominator variable?
Do not invert a sum term by term.
Answer
Isolate its reciprocal, combine the other fractions, then take reciprocals with domain restrictions.
Example
From \(1/f=1/u+1/v\), isolate \(1/v\) first.
Card 18 of 20
MediumStudy card 18
Prompt
What common error occurs after factoring \(x(a+b)=M\)?
The parentheses form one coefficient.
Answer
Dividing by only \(a\) or \(b\) instead of the complete factor \(a+b\).
Example
The correct result is \(x=M/(a+b)\).
Card 19 of 20
HardStudy card 19
Prompt
What is a reliable workflow for a complex literal equation?
Keep target-containing structure visible.
Answer
Name the target, simplify, collect target terms, factor if needed, divide by the full coefficient, and audit restrictions.
Example
Use this on \(A=p(x+r)-q(x-s)\).
Card 20 of 20
HardStudy card 20
Prompt
How can you audit a rearranged formula?
Check grouping, signs, and units.
Answer
Reverse the operations or substitute the rearranged target expression into the original structure and simplify.
Example
Substituting \(t=d/r\) into \(d=rt\) returns \(d=d\).
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