Rearranging literal equations
A literal equation is one with several letters, such as a formula. Rearranging a formula means solving for one chosen letter while treating the others as if they were known constants. The moves are identical to ordinary solving — inverse operations, both sides — you just carry letters instead of numbers and stop when your target letter stands alone.
Solve the area formula A = (1/2) b h for h. Multiply both sides by 2: 2A = b h Divide both sides by b: 2A / b = h So h = 2A / b. Check the logic: if A = 10 and b = 4, then h = 20/4 = 5, and (1/2)(4)(5) = 10 = A. Consistent.
Translating a word problem
A word problem is solved by translation, not magic. Name the unknown with a letter, write the relationship the words describe as an equation, solve it, then interpret the answer back in the original units and check it against the story.
- Read the question and decide exactly what the unknown is; name it (let x = …).
- Translate each relationship into algebra: 'is' becomes =, 'more than' becomes +, 'of' often becomes ×.
- Solve the equation by isolating the variable.
- Interpret the number in context and check that it answers the actual question.
Word problem: A shirt costs $8 more than twice the price of a cap.
Together they cost $50. Find the price of the cap.
Let c = price of the cap (in dollars).
Shirt = 2c + 8.
Together: c + (2c + 8) = 50
3c + 8 = 50
3c = 42
c = 14
Cap = $14, shirt = 2(14) + 8 = $36, and 14 + 36 = 50. Story checks out.Identities, contradictions, and clean answers
Not every linear equation has exactly one solution. If solving collapses the variable and leaves something always true, like 5 = 5, the equation is an identity and every number is a solution. If it leaves something always false, like 5 = 8, it is a contradiction with no solution. Both outcomes are honest answers, not mistakes — recognizing them keeps you from chasing an x that does not exist.
Identity: 2(x + 3) = 2x + 6 2x + 6 = 2x + 6 -> 0 = 0 (always true) Solution set: all real numbers, (-infinity, infinity). Contradiction: x + 1 = x + 4 1 = 4 (never true) Solution set: empty set, no solution.