Solve for a chosen symbol while keeping units and zero-coefficient cases visible.
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Try
A rectangle has area A and width w. You need its length l. Start with A = lw. Explain why l = A/w needs more than a rearranged string of symbols.
Predict
If A = 30 square metres and w = 5 metres, predict l and its unit. Dividing square metres by metres gives metres: l = 6 m. What goes wrong if w = 0?
Experiment
Make a table for A = 30 with widths 2, 3 and 5 metres. The lengths are 15, 10 and 6 metres. Their products all equal 30. As width decreases, the required length increases. Do not put a zero-width row through the division formula; the original equation would say 30 = 0, which is impossible.
Understand
A literal formula is an equation with several symbols. When solving for one, treat the others as fixed quantities while retaining their conditions. In A = lw, division by w is valid when w is nonzero. Physical rectangles normally also require positive dimensions.
For perimeter P = 2l + 2w, subtract 2w and divide by 2: l = (P − 2w)/2 = P/2 − w. If P = 26 m and w = 4 m, l = 9 m. A value of w larger than P/2 would yield a negative l, signaling a physically impossible specification.
For A = h(a + b)/2, multiply by 2 before isolating the grouped sum: 2A = h(a + b). With h nonzero, divide to obtain 2A/h = a + b, then b = 2A/h − a. Dividing only one term in a sum would not preserve the equation.
Exceptional cases should return to the original. In y = mx + b, m = 0 gives y = b. If that equality holds, x is unrestricted; if it does not, no x can repair it. The expression (y − b)/m cannot represent these cases because division by zero is undefined.
Apply
Work the numeric and symbolic questions below. For each rearrangement, write the forbidden divisor values. For each contextual answer, track units and check the original formula. The written response requires separate cases, not a decimal approximation.
Continue
Absolute value introduces a different kind of case split: a distance can arise from a position on either side of a reference point. Carry forward the practice of identifying conditions before manipulating a formula.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
F = (9/5)C + 32. What is C when F = 14?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A = h(a + b)/2. If A = 42 cm², h = 6 cm and a = 5 cm, find b in cm.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For nonzero r, which formula isolates t in P = Q(1 + rt), assuming Q is nonzero?
Distance d = vt. A vehicle travels 45 km at constant 30 km/hour. Find t in hours exactly.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Solve y = mx + b for x, including m = 0. Explain why x = (y − b)/m is not the whole answer.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
