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📝 How to Solve a Formula for a Specific Variable (14 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 14 questions available

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What is How to Solve a Formula for a Specific Variable?

Definition:
Solving a formula for a specific variable means rearranging the equation to isolate that variable on one side, treating all other variables as constants. This is done using the same properties of equality, and the result expresses the chosen variable in terms of the others.

Working:
For $A = lw$ (area of a rectangle), to solve for $w$ , divide both sides by $l$ : $w = A/l$ . For $P = 2l + 2w$ (perimeter), to solve for $l$ , subtract $2w$ : $P - 2w = 2l$ , then divide by 2: $l = (P - 2w)/2$ .

Example:
Solve $V = \frac{1}{3}Bh$ for $B$ (volume of a pyramid). Multiply both sides by 3: $3V = Bh$ , then divide by $h$ : $B = 3V/h$ .

Reason:
This skill is essential in science and geometry to calculate unknown quantities from known ones without recalculating the entire formula.

Easy

Medium

Hard

📝 All How to Solve a Formula for a Specific Variable MCQs

Q1. Which equation correctly solves $A=lw$ for $w$ while keeping the equation equivalent for every valid value of $l$ ?

w=A-l

w=\frac{A}{l}

✅

w=Al

w=\frac{l}{A}

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: To isolate $w$ , divide both sides of $A=lw$ by $l$ . This gives $w=\frac{A}{l}$ , provided $l\neq0$ . The other choices either subtract or multiply instead of applying the inverse operation needed to isolate the variable.

Q2. The formula $P=2l+2w$ is used to determine the width of a rectangle. Which rearrangement isolates $w$ most efficiently?

w=\frac{P-2l}{2}

✅

w=P-2l

w=\frac{P}{2}-2l

w=2(P-2l)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Subtracting $2l$ from both sides gives $P-2l=2w$ . Dividing by $2$ then produces $w=\frac{P-2l}{2}$ . This sequence preserves equivalence and directly isolates the requested variable without changing the coefficient incorrectly.

Q3. A student wants to solve $y=3x-7$ for $x$ . Which expression correctly represents $x$ , and why is simply adding $7$ to $y$ incomplete?

x=\frac{y+7}{3}

, because the coefficient

3

must also be removed ✅

x=3y+7

, because multiplication comes first

x=\frac{y-7}{3}

, because the constant remains negative

x=y+7-3

, because all operations can be reversed together

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Adding $7$ first gives $y+7=3x$ , but $x$ is still multiplied by $3$ . Dividing by $3$ completes the isolation, yielding $x=\frac{y+7}{3}$ . The distractors reflect common sign and operation-order errors.

Q4. A science formula is $F=\frac{9}{5}C+32$ . A student rearranges it as $C=\frac{5F}{9}+32$ . What is the best diagnosis of the error?

A.The student divided the entire expression by

9/5

incorrectly and failed to remove

32

first ✅

B.The student should multiply

F

32

before dividing

C.The student correctly isolated

C

but used an unnecessary form

D.The student only needs to change

5

into

9

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The constant $32$ must first be subtracted from both sides, producing $F-32=\frac{9}{5}C$ . Multiplying by $\frac{5}{9}$ then gives $C=\frac{5}{9}(F-32)$ . The student's expression incorrectly leaves $32$ outside the transformation.

Q5. A rectangle has area $A=lw$ . Two students solve for $l$ . Student 1 writes $l=\frac{A}{w}$ , while Student 2 writes $l=Aw$ . Which comparison is correct?

A.Student 1 is correct because division by

w

cancels multiplication by

w

✅

B.Student 2 is correct because multiplying preserves the original formula

C.Both are correct for positive

w

D.Neither is correct because

A

cannot be divided

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Since $A=lw$ , dividing both sides by $w$ gives $\frac{A}{w}=l$ . Multiplying by $w$ would instead produce $Aw=lw^2$ , which is not equivalent to the original equation. Therefore Student 1 correctly isolates $l$ .

Q6. A construction worker uses $V=\pi r^2h$ and needs a formula for $h$ . Which expression is most useful when the volume and radius are known?

h=\frac{V}{\pi r^2}

✅

h=V\pi r^2

h=\frac{\pi r^2}{V}

h=\frac{V}{\pi}+r^2

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The height $h$ is multiplied by $\pi r^2$ . Dividing both sides by that entire factor isolates $h$ , giving $h=\frac{V}{\pi r^2}$ . Keeping $\pi r^2$ together is essential because both factors multiply $h$ .

Q7. A taxi fare is modeled by $C=b+mr$ , where $b$ is the starting fee, $m$ is the cost per kilometer, and $r$ is distance. If the company wants a formula for $r$ , which is correct?

r=\frac{C-b}{m}

✅

r=\frac{C+b}{m}

r=m(C-b)

r=\frac{b-C}{m}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Subtracting the starting fee $b$ from the total cost leaves $mr$ . Dividing by the rate $m$ then isolates distance, producing $r=\frac{C-b}{m}$ . This rearrangement also matches the practical interpretation that variable cost equals total cost minus base cost.

Q8. A student solves $S=ut+\frac12at^2$ for $u$ and writes $u=\frac{S}{t}+\frac12at$ . Is this equivalent to the original formula?

A.Yes, because every term was divided by

t

B.No, because the entire expression

S-\frac12at^2

must be divided by

t

✅

C.No, because

u

should equal

S-\frac12at^2

D.Yes, because

\frac12at^2

becomes

\frac12at

after division

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: To isolate $u$ , first subtract $\frac12at^2$ from both sides: $S-\frac12at^2=ut$ . Dividing the entire left side by $t$ gives $u=\frac{S-\frac12at^2}{t}$ . The student's method incorrectly divides only selected terms.

Q9. For $y=mx+b$ , a graph shows several lines with different slopes and intercepts. A student wants an expression for $m$ using a known point $(x,y)$ and known intercept $b$ . Which relationship should the student use?

m=\frac{y-b}{x}

✅

m=\frac{x-b}{y}

m=yx+b

m=\frac{y+b}{x}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Starting with $y=mx+b$ , subtract $b$ to obtain $y-b=mx$ . Dividing by $x$ gives $m=\frac{y-b}{x}$ , assuming $x\neq0$ . This relationship also shows how the vertical difference from the intercept determines the slope for a chosen point.

Q10. A company models total production cost with $C=F+qn$ , where $F$ is fixed cost, $q$ is cost per item, and $n$ is the number of items. If $q$ must be determined from measured $C,F,n$ , what formula should be used?

q=\frac{C-F}{n}

✅

q=\frac{C+F}{n}

q=n(C-F)

q=\frac{F-C}{n}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Subtracting fixed cost from total cost isolates the portion attributable to producing $n$ items: $C-F=qn$ . Dividing by $n$ then gives $q=\frac{C-F}{n}$ . This formula directly separates fixed expenses from variable production expenses.

Q11. Two students rearrange $K=\frac12mv^2$ for $m$ . Student A writes $m=\frac{2K}{v^2}$ . Student B writes $m=2Kv^2$ . Which reasoning best evaluates their answers?

A.A is correct because multiplying by

2

and dividing by

v^2

isolates

m

✅

B.B is correct because

v^2

must be moved by multiplication

C.Both are equivalent when

v\neq0

D.A is incorrect because

m=\frac{K}{2v^2}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Multiplying $K=\frac12mv^2$ by $2$ gives $2K=mv^2$ . Dividing by $v^2$ isolates $m$ , resulting in $m=\frac{2K}{v^2}$ . Student B reverses the required operation on $v^2$ , producing a nonequivalent expression.

Q12. Suppose $d=rt$ is used to model travel. A driver knows distance $d$ and speed $r$ , but wants travel time $t$ . Which interpretation correctly explains the rearrangement?

A.Time equals distance divided by speed, so

t=\frac{d}{r}

✅

B.Time equals speed divided by distance, so

t=\frac{r}{d}

C.Time equals distance multiplied by speed, so

t=dr

D.Time equals the difference between distance and speed, so

t=d-r

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: In $d=rt$ , speed $r$ and time $t$ multiply to produce distance. Dividing both sides by speed removes $r$ , giving $t=\frac{d}{r}$ . The result also agrees with the real-world meaning of travel time.

Q13. A challenging algebra check starts with $M=\frac{a+b}{c}$ . A student claims $a=Mc-b$ . Which verification most strongly supports the claim?

A.Substituting

a=Mc-b

gives

M=\frac{Mc-b+b}{c}=M

, so the rearrangement is valid when

c\neq0

✅

B.Substituting gives

M=\frac{Mc+b}{c}

, so the claim is always valid

C.The claim is valid only when

b=0

D.The claim cannot be checked because

M

is not a variable

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Multiplying the original equation by $c$ gives $Mc=a+b$ . Subtracting $b$ yields $a=Mc-b$ . Substitution confirms the result because the numerator becomes $Mc$ , which divided by nonzero $c$ returns $M$ .

Q14. A model uses $R=\frac{P}{1+kt}$ . A researcher needs to solve for $t$ before estimating how long a process has operated. Which expression is correct?

t=\frac{P/R-1}{k}

✅

t=\frac{R/P-1}{k}

t=\frac{P-R}{1+k}

t=\frac{P}{R+k}-1

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Starting with $R=\frac{P}{1+kt}$ , multiply by $1+kt$ to obtain $R(1+kt)=P$ . Dividing by $R$ gives $1+kt=P/R$ , and subtracting $1$ before dividing by $k$ yields $t=\frac{P/R-1}{k}$ .

🔗 Related Topics (MCQs)

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