Key takeaway: A literal equation is a formula-style statement full of letters, and you rework it with inverse operations until one chosen letter stands alone.
Grade 9 – Algebra 1A-CED.A.4Expressions and Equations
Definition of Literal Equation
A literal equation mixes several letters in one relationship, like the formulas in geometry or physics class. Instead of plugging in numbers right away, you solve it for one variable and keep all the other letters in the answer.
The rules stay the same as for numbers. Peel off the added or subtracted pieces, then the multipliers or divisors, always doing the same step to both sides. Treat every other letter like a number. Rearranging a formula once saves time when you must use it again and again with different values, for instance when a carpenter needs a missing side length for many different boards.
Formula and notation
In a literal equation the chosen variable is named in the instruction, for example “solve for \(a\)”. The final line has that variable alone on one side:
\[A = \frac{1}{2}h(a + b) \ \Longrightarrow\ a = \frac{2A}{h} – b\]
Fractions with letters are fine; just keep the denominator nonzero.
Key facts
- The target variable must end up alone on one side, with no other letters beside it.
- Inverse operations work the same as with numbers, in reverse order of operations.
- A literal equation can be rearranged first and evaluated later, with the same result.
- Check the rearranged formula by plugging in a simple set of numbers.
Literal Equation example
A trapezoid-shaped garden bed has area \(A = \frac{1}{2}h(a + b)\), where \(a\) and \(b\) are the parallel sides. Rearrange the rule for \(a\), then compute \(a\) when \(A = 96\), \(h = 8\) and \(b = 7\).
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Multiply both sides by 2: \(2A = h(a + b)\).
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Divide both sides by \(h\): \(\frac{2A}{h} = a + b\). Subtract \(b\): \(a = \frac{2A}{h} – b\).
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Substitute: \(a = \frac{2(96)}{8} – 7 = 24 – 7 = 17\).
The formula is \(a = \frac{2A}{h} – b\), and \(a = 17\) for these values.
Common mistake
Watch out
After reaching \(\frac{2A}{h} = a + b\), Gabe writes \(a = \frac{2A}{h} + b\).
Why it is wrong
To remove \(+b\) from the right side, subtract \(b\), do not add it. Check with numbers: \(24 + 7 = 31\) gives the wrong side length, while \(24 – 7 = 17\) satisfies \(\frac{1}{2}(8)(17 + 7) = 96\).
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Expressions and Equations