Algebra Practice

Solving for a Variable Practice Test

Rearrange formulas with inverse operations, fractions, powers, roots, and several symbolic factors.

Solving for a Variable Practice Test

This test has 20 questions

Instant feedback · Worked explanations
Target variable selected

After the test · formula rearrangement lab

Solving for a variable means rewriting a formula so the chosen symbol stands alone while the relationship stays equivalent

In literal equations, several letters may appear at once, but only one is the current target. Treat the other letters as known quantities. The algebra is the same balance logic used in ordinary equations: undo operations in a controlled order until the target variable is isolated.

Literal equationsFormula rearrangementInverse operations Grouped factorsFractionsGeometry formulas Rate formulasRestrictions
First circle the variable you are solving for. Many rearrangement mistakes happen because students start manipulating the formula before fixing a clear target.
Original formula
A = lw
Solve for l
l = A/w
Solve for w
w = A/l

Rearrangement conveyor

Work backward through the operations that are attached to the target variable

Target Identify the variable.

Example: solve A = ½bh for h.

Undo outer factor Multiply by 2.

2A = bh.

Expose target h is multiplied by b.

Only one factor remains attached to h.

Undo coefficient Divide by b.

2A/b = h.

Present h = 2A/b

Write the target variable clearly on one side.

Formula rack

A single algebra skill unlocks many familiar formulas

Rectangle area
A = lw
w = A/l
Ohm's law
V = IR
R = V/I
Linear form
y = mx + b
x = (y − b)/m
Trapezoid area
A = ½h(b₁ + b₂)
h = 2A/(b₁ + b₂)
Rearrangement station · solve P = 2l + 2w for w
1
P = 2l + 2w The target w is inside the term 2w.
2
P − 2l = 2w Subtract 2l from both sides.
3
(P − 2l)/2 = w Divide the entire left side by 2.
4
w = P/2 − l This is an equivalent simplified form.

Grouping gate

If the target multiplies an entire grouped expression, divide by that complete group

Start
A = ½h(b₁ + b₂) Target: h.
Remove ½
2A = h(b₁ + b₂) The whole quantity b₁ + b₂ multiplies h.
Divide by group
h = 2A/(b₁ + b₂) Keep parentheses around the denominator sum.

Formula families

Different formula structures suggest different isolation moves

Product formula

V = IR, solve for I

I = V/R

Undo multiplication by R with division.

Additive formula

y = mx + b, solve for b

b = y − mx

Subtract mx from both sides.

Fraction formula

d = n/t, solve for n

n = dt

Multiply both sides by t.

Temperature formula

F = (9/5)C + 32, solve for C

C = (5/9)(F − 32)

Subtract 32 first, then multiply by 5/9.

A = lw → w = A/l For the rearranged division to be defined, l ≠ 0.
V = IR → R = V/I This algebraic form requires I ≠ 0.
A = ½h(b₁ + b₂) → h = 2A/(b₁ + b₂) The rearranged form requires b₁ + b₂ ≠ 0.

Unit consistency check

Units provide a second way to test whether a rearranged formula is plausible

Formula d = rt
Solve for r r = d/t
Units distance ÷ time
Interpretation rate units are recovered correctly

Equivalence switch

Test a rearrangement by substituting numbers into both the original and rearranged formula

Original:
A = lw
A = 30, l = 5
30 = 5w
Rearranged:
w = A/l
w = 30/5
w = 6
Substituting w = 6 back into A = lw gives 30 = 5·6, so the rearranged formula is consistent with the original relationship.

Rearrangement error routing

Most mistakes come from dividing or subtracting only part of a grouped expression

Partial division
P = 2l + 2w → w = P − 2l/2 w = (P − 2l)/2

The entire difference P − 2l must be divided by 2.

Lost grouping
A = ½h(b₁ + b₂) → h = 2A/b₁ + b₂ h = 2A/(b₁ + b₂)

The whole sum b₁ + b₂ is a factor of h.

Wrong order
F = (9/5)C + 32 → C = (5/9)F − 32 C = (5/9)(F − 32)

Subtract 32 before undoing multiplication by 9/5.

Restriction lost
A = lw → w = A/l with no condition w = A/l, with l ≠ 0

Division by the isolating factor requires that factor to be nonzero.

Rearrangement constellation

Use missed questions to identify which part of formula isolation needs review

Literal-equation errors are easier to fix when separated into target choice, operation order, grouping, and validation.

Target selection The wrong variable was treated as the unknown.
Inverse order Operations were undone in the wrong sequence.
Grouping A sum, difference, or factor was divided only partially.
Restrictions & checks A zero-denominator condition or verification step was missed.