A literal equation contains several letters. Solving for one means isolating it exactly as you would isolate x, treating the other letters as numbers. Units multiply and divide alongside the numbers, and a wrong final unit means a wrong setup.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A rectangle’s perimeter is P = 2l + 2w. One rectangle has perimeter 26 cm and length 8 cm. What is its width?
cm
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Treat other letters as numbers
A = lw for l: divide by w, l = A/w. P = 2l + 2w for w: subtract 2l, P − 2l = 2w, divide by 2, w = (P − 2l)/2. The fact that they are letters changes nothing about the method.
A harder one
F = (9/5)C + 32 for C: subtract 32, F − 32 = (9/5)C, multiply by 5/9, C = (5/9)(F − 32). Check: F = 212 gives C = (5/9)(180) = 100.
Why it is useful
Rearranging once lets you answer many questions. t = d/r answers every time question in one step: 300 miles at 60 mph is 5 hours; 150 at 50 is 3. No re-solving each time.
Units carry information
60 miles/hour × 3 hours = 180 miles: the hours cancel and miles remain. Treat units as factors that multiply and divide. When the numbers are right, the units come out right too.
Units as a check
If a calculation for time produces miles², the setup is wrong. This catches errors that arithmetic checking misses entirely, because the arithmetic can be perfect on the wrong formula.
Converting units
72 km/h to m/s: 72 km/h × (1000 m/km) × (1 h/3600 s) = 20 m/s. Multiply by fractions equal to 1, arranged so the unwanted units cancel. Report a walking distance in km, not mm.
Step 2: Try It Yourself
Tap and try it out.
Rearrange F = (9/5)C + 32 for C, then convert 72 km/h to m/s, checking the units
- 1F − 32 = (9/5)C− 32subtract 32 from both sides
Step 0 of 4
Area · squares inside
24
Perimeter · walk around
20
4 × 6 = 24 · 4 + 6 + 4 + 6 = 20
Area counts the squares inside. Perimeter measures the walk all the way round.
Step 3: In Real Life
Speed, distance, time
d = rt is written for distance. A driver wants time: t = d/r. Miles over miles per hour leaves hours, and the units confirm the rearrangement was right.
Step 4: Watch an Example
One step at a time.
Watch Ravi Rearrange a Taxi Fare
A taxi charges $3 to start plus $2 a mile, so the fare is C = 3 + 2m dollars for m miles. Ravi wants a formula for the miles.
- Step 1
Ravi wants m alone, and he treats C as if it were a number he cannot see. The 3 is added to 2m, so it comes off first: taking 3 from both sides leaves C − 3 = 2m.
Watch Leo Rearrange a Plumber’s Bill
A plumber charges a $40 service call fee plus $25 an hour: C = 40 + 25h dollars for h hours. Leo writes the first line, and you write the rest.
- Step 1
Leo wants h alone. The 40 was added last, so taking 40 from both sides leaves C − 40 = 25h.
Step 5: Your Turn
Practice makes it stick.
Filling the Pool
Problem 1 of 17
A pool holds 1,200 gallons. A hose fills it at 8 gallons per minute. How many hours does it take to fill, as a decimal?
hours
The Rental Car
Problem 2 of 17
A car rental costs $25 a day plus $0.20 a mile: C = 25d + 0.2m dollars. Maya rents for 3 days and the bill is $105. How many miles did she drive?
miles
Solve for the Letter
Treat every other letter as a number. Undo the last thing done to your letter first.
Problem 3 of 17
A square’s perimeter is P = 4s. Solve for s.
Problem 4 of 17
Solve 2x + y = 10 for y.
Problem 5 of 17
A triangle with two equal sides a and a third side b has perimeter P = 2a + b. Solve for a.
Problem 6 of 17
Rearrange d = rt for r, then find the speed for a trip of 150 miles in 3 hours.
miles per hour
Problem 7 of 17
Convert 30 meters per minute to meters per second.
m/s
Whose Step?
One line in each piece of working is a slip. Find it, then fix it.
Problem 8 of 17
Priya buys 3 books at b dollars each and 6 pens at c dollars each, so she pays A = 3b + 6c. She rearranged it for b. One line is a slip. Which?
Priya’s rearrangement of A = 3b + 6c for b
- 1A = 3b + 6c
- 2A − 6c = 3b− 6c
- 3A/3 − 6c = b÷ 3
- Answerb = A/3 − 6c
Problem 9 of 17
What should Priya’s formula for b be? Type it.
Problem 10 of 17
Dev wants the time for a 240-mile drive at 60 miles per hour. One line is a slip. Which?
Dev’s time for 240 miles at 60 mph
- 1d = rt
- 2240 = 60t
- 3t = 240 × 60÷ 60
- Answert = 14,400 hours
Problem 11 of 17
How long does Dev’s 240-mile drive at 60 miles per hour really take?
hours
Problem 12 of 17
Noor converted 18 km/h to meters per second. One line is a slip. Which?
Noor’s conversion of 18 km/h to m/s
- 118 km/h
- 2= 18,000 m/h× 1000 m/km
- Answer= 64,800,000 m/s× 3600 s/h
Problem 13 of 17
What is 18 km/h in meters per second?
m/s
Pick the Move
Decide the move before you make it.
Problem 14 of 17
To solve 5k + j = m for k, which move comes first?
Problem 15 of 17
Ana wants 90 minutes in hours. She multiplies 90 min by a fraction equal to 1. Which fraction?
Same Thing?
Decide without starting over. A test number is enough.
Problem 16 of 17
Select every formula that says the same thing as C = (5/9)(F − 32).
Problem 17 of 17
Select every one that writes 3 km in meters.
Step 6: Quick Check
Show what you know.
Question 1 of 2
A = lw with A = 42 and w = 7. What is l?
Question 2 of 2
How do units help check a calculation?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Driving at 60 miles per hour with one half-hour stop, a trip of d miles takes t = d/60 + 0.5 hours. Solve for d, so the formula tells you how far you can go in t hours.
Stretch 2 of 2
Every speed in km/h can be turned into m/s by dividing it by one fixed number. What is that number?
What You Learned
- Solve a literal equation by isolating the wanted letter, treating others as numbers.
- Rearrange once, answer many questions.
- Units multiply and divide alongside the numbers, and a wrong final unit means a wrong setup.
- Convert units by multiplying by fractions equal to 1.
For the grown-up
The misconception this lesson targets is that letters in a formula behave differently from numbers, so the careful moves of an ordinary equation stop applying. It shows most as dividing only one term of a side, A/3 − 6c instead of (A − 6c)/3: with numbers nobody would halve 10 − 4 by halving only the 10, but with letters the side stops looking like one thing. The second slip treats units as labels rather than factors, so a rate gets multiplied where it should divide, or a conversion fraction goes in upside down. Carrying the units through every line catches both, because a time that comes out in miles² per hour or a speed in m·s/h² is a setup error the arithmetic alone will never show. A test number, such as F = 212 with C = 100, catches a slip in a rearranged formula the same way.