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📝 Solving formulas for a variable (13 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 13 questions available

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What is Solving formulas for a variable?

Definition:
Solving formulas for a variable involves isolating a specific variable in an equation that contains multiple variables. This uses inverse operations (addition/subtraction, multiplication/division) to get the desired variable alone on one side, treating other variables as constants.

Working:
For $F = \frac{9}{5}C + 32$ (Fahrenheit to Celsius), solve for $C$ : subtract 32: $F - 32 = \frac{9}{5}C$ , multiply by $5/9$ : $C = \frac{5}{9}(F - 32)$ .

Example:
Solve $A = \frac{1}{2}bh$ for $h$ (area of triangle). Multiply by 2: $2A = bh$ , divide by $b$ : $h = 2A/b$ .

Reason:
This is a critical skill in STEM fields, allowing conversion between different forms of formulas and solving for any parameter.

Easy

Medium

Hard

📝 All Solving formulas for a variable MCQs

Q1. Given $A=lw$ , which expression correctly isolates $w$ when $A$ and $l$ are known?

w=A-l

w=Al

w=A/l

✅

w=l/A

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: To isolate $w$ , divide both sides of $A=lw$ by $l$ , assuming $l\neq0$ . This gives $w=A/l$ . The other choices result from subtracting or multiplying instead of applying the inverse operation correctly.

Q2. The formula $P=2l+2w$ represents a rectangle's perimeter. Which form isolates $l$ ?

l=P-2w

l=(P-2w)/2

✅

l=(P+2w)/2

l=P/(2w)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Starting with $P=2l+2w$ , subtract $2w$ from both sides to obtain $P-2w=2l$ . Dividing by 2 then gives $l=(P-2w)/2$ , preserving equivalence at every step.

Q3. A formula is $C=5(F-32)/9$ . Which sequence correctly isolates $F$ without changing the relationship?

A.Multiply by 9, divide by 5, then add 32. ✅

B.Add 32, multiply by 9, then divide by 5.

C.Subtract 32, multiply by 5, then divide by 9.

D.Multiply by 5, divide by 9, then subtract 32.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: To isolate $F$ , first multiply by 9, giving $9C=5(F-32)$ . Divide by 5 to obtain $9C/5=F-32$ , then add 32. This produces $F=9C/5+32$ .

Q4. A physics model uses $v=u+at$ . A student wants $t$ and writes $t=(v-u)a$ . Why is this incorrect?

A.Acceleration must be added instead of multiplied.

B.The entire difference

v-u

must be divided by

a

. ✅

C.Initial velocity must be divided by

a

first.

D.Velocity must be multiplied by acceleration.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Subtracting $u$ from both sides gives $v-u=at$ . Since $a$ multiplies $t$ , division by $a$ is required to isolate $t$ . Therefore, $t=(v-u)/a$ , assuming $a\neq0$ .

Q5. A rectangle has area 84 square units and width 7 units. Using $A=lw$ , which expression and value correctly determine its length?

l=84-7=77

l=84+7=91

l=84/7=12

✅

l=7/84=1/12

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Because area equals length times width, isolate length by dividing the area by the width. Substituting $A=84$ and $w=7$ gives $l=84/7=12$ . Subtraction or reversing the division does not preserve the formula.

Q6. A phone plan is modeled by $T=25+0.08m$ , where $T$ is total cost and $m$ is the number of messages. Which expression isolates $m$ ?

m=(T+25)/0.08

m=(T-25)/0.08

✅

m=0.08(T-25)

m=25-0.08T

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Subtracting the fixed charge 25 from both sides gives $T-25=0.08m$ . Dividing by 0.08 isolates $m$ . Thus $m=(T-25)/0.08$ , which correctly separates the variable-dependent charge from the fixed charge.

Q7. A student transforms $S=2\pi rh+2\pi r^2$ into $h=(S-2\pi r^2)/(2\pi r)$ . Which reasoning justifies the result?

A.Both terms are divided by

2\pi r

after the second term is subtracted. ✅

B.The radius can be canceled from only the second term.

C.The entire equation is first multiplied by

2\pi r

D.The

2\pi r^2

term is ignored because it contains no

h

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: To isolate $h$ , subtract $2\pi r^2$ from both sides, giving $S-2\pi r^2=2\pi rh$ . Dividing the entire equation by $2\pi r$ then yields the stated expression, assuming $r\neq0$ .

Q8. The equation $y=mx+b$ can be viewed as a line. If $y$ and $b$ are fixed and $m\neq0$ , which rearrangement gives the corresponding $x$ -coordinate?

x=(y-b)/m

✅

x=(y+b)/m

x=m(y-b)

x=(b-y)/m

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Subtracting the intercept $b$ from both sides gives $y-b=mx$ . Dividing by the slope $m$ isolates $x$ , producing $x=(y-b)/m$ . This also represents how a known vertical value determines the horizontal coordinate.

Q9. On a graph of $y=4x-12$ , a point has $y=20$ . Which rearrangement and calculation correctly find its $x$ -coordinate?

x=(20+12)/4=8

✅

x=(20-12)/4=2

x=4(20-12)=32

x=(12-20)/4=-2

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substituting $y=20$ into $y=4x-12$ gives $20=4x-12$ . Adding 12 produces $32=4x$ , and dividing by 4 gives $x=8$ . The point therefore has horizontal coordinate 8.

Q10. Two students isolate $x$ from $ax+c=b$ . Student 1 writes $x=(b-c)/a$ . Student 2 writes $x=b/a-c$ . Assuming $a\neq0$ , which statement is most accurate?

A.Only Student 2 is always correct.

B.Both are always equivalent.

C.Student 1 is correct because the numerator must be divided as a whole by

a

. ✅

D.Student 2 is correct because division distributes over subtraction.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Student 1 correctly subtracts $c$ first, obtaining $ax=b-c$ , then divides the entire right side by $a$ . Student 2's expression generally means $b/a-c$ , which does not equal $(b-c)/a$ unless special values make them coincide.

Q11. A travel formula is $d=rt$ . A driver knows the distance $d=315$ km and time $t=4.5$ hours. Which method correctly isolates and calculates the rate?

A.Multiply 315 by 4.5, giving 1417.5 km/h.

B.Divide 315 by 4.5, giving 70 km/h. ✅

C.Divide 4.5 by 315, giving approximately 0.014 km/h.

D.Subtract 4.5 from 315, giving 310.5 km/h.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: From $d=rt$ , divide both sides by $t$ to isolate $r$ , giving $r=d/t$ . Substituting the measurements gives $315/4.5=70$ , so the driver's average rate is 70 km/h.

Q12. A quantity is defined by $Q=(x+6)/4$ . Another student claims the isolated form is $x=4Q-6$ . How can the claim be verified most efficiently?

A.Substitute

x=4Q-6

into the original formula and simplify. ✅

B.Multiply the original formula by 6 only.

C.Compare the coefficients of

x

without substitution.

D.Subtract

Q

from both sides of the original equation.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Starting with $Q=(x+6)/4$ , multiplying both sides by 4 gives $4Q=x+6$ , and subtracting 6 gives $x=4Q-6$ . Substitution into the original formula returns $Q$ , confirming the rearrangement.

Q13. A model is $E=mc^2$ . If $m$ is the variable to isolate, which expression is correct, and what key condition is required?

m=Ec^2

, with

c=0

m=E/c^2

, with

c\neq0

✅

m=c^2/E

, with

E\neq0

m=E-c^2

, with no restriction

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Since $c^2$ multiplies $m$ , dividing both sides by $c^2$ gives $m=E/c^2$ . Division requires $c^2\neq0$ , so $c\neq0$ . The other choices either reverse the division or use an invalid operation.

🔗 Related Topics (MCQs)

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