In this lesson you’ll learn how to rearrange formulas to solve for any variable, not just x.
A literal equation has multiple variables (e.g., d = rt for distance = rate × time).
To solve for one variable, use inverse operations to isolate it. Treat other letters as constants.
Example: Solve for t in d = rt
d = r t Divide both sides by r d r = t \begin{array}{l|rcl}
& d &=& rt \\
\text{Divide both sides by r} & \dfrac{d}{r} &=& t
\end{array} Divide both sides by r d r d = = r t t
More complex: Solve for h in V = 1 3 π r 2 h V = \tfrac{1}{3}\pi r^2 h V = 3 1 π r 2 h
Cone — V = ⅓πr²h h r
V = 1 3 π r 2 h Multiply both sides by 3 3 V = π r 2 h Divide both sides by π r 2 3 V π r 2 = h \begin{array}{l|rcl}
& V &=& \tfrac{1}{3}\pi r^2 h \\
\text{Multiply both sides by 3} & 3V &=& \pi r^2 h \\
\text{Divide both sides by } \pi r^2 & \dfrac{3V}{\pi r^2} &=& h
\end{array} Multiply both sides by 3 Divide both sides by π r 2 V 3 V π r 2 3 V = = = 3 1 π r 2 h π r 2 h h
Steps are the same as regular equations, just keep other variables attached.
Solve for r in C = 2πr (circumference of circle)
Circle — C = 2πr r C
C = 2 π r Divide both sides by 2 π C 2 π = r \begin{array}{l|rcl}
& C &=& 2\pi r \\
\text{Divide both sides by } 2\pi & \dfrac{C}{2\pi} &=& r
\end{array} Divide both sides by 2 π C 2 π C = = 2 π r r
Solve for m in y = mx + b (slope-intercept form)
y = m x + b Subtract b from both sides y − b = m x Divide both sides by x y − b x = m \begin{array}{l|rcl}
& y &=& mx + b \\
\text{Subtract b from both sides} & y - b &=& mx \\
\text{Divide both sides by x} & \dfrac{y - b}{x} &=& m
\end{array} Subtract b from both sides Divide both sides by x y y − b x y − b = = = m x + b m x m
Solve for t in A = P(1 + rt) (simple interest)
A = P ( 1 + r t ) Distribute A = P + P r t Subtract P from both sides A − P = P r t Divide both sides by Pr A − P P r = t \begin{array}{l|rcl}
& A &=& P(1 + rt) \\
\text{Distribute} & A &=& P + Prt \\
\text{Subtract P from both sides} & A - P &=& Prt \\
\text{Divide both sides by Pr} & \dfrac{A - P}{Pr} &=& t
\end{array} Distribute Subtract P from both sides Divide both sides by Pr A A A − P P r A − P = = = = P ( 1 + r t ) P + P r t P r t t
Rearranging formulas is used constantly:
How long to drive? - You need to drive 300 miles at 60 mph. How long will it take?
d = r t Divide both sides by r d r = t Plug in values 300 60 = 5 hours \begin{array}{l|rcl}
& d &=& rt \\
\text{Divide both sides by r} & \dfrac{d}{r} &=& t \\
\text{Plug in values} & \dfrac{300}{60} &=& 5 \text{ hours}
\end{array} Divide both sides by r Plug in values d r d 60 300 = = = r t t 5 hours
It will take 5 hours.
How fast were you going? - You drove 240 miles in 4 hours. What was your speed?
d = r t Divide both sides by t d t = r Plug in values 240 4 = 60 mph \begin{array}{l|rcl}
& d &=& rt \\
\text{Divide both sides by t} & \dfrac{d}{t} &=& r \\
\text{Plug in values} & \dfrac{240}{4} &=& 60 \text{ mph}
\end{array} Divide both sides by t Plug in values d t d 4 240 = = = r t r 60 mph
Your speed was 60 mph.
How much to invest? - You want 1200 dollars after 3 years at 5% simple interest. How much do you need to start with?
A = P ( 1 + r t ) Divide both sides by ( 1 + r t ) A 1 + r t = P Plug in values 1200 1 + ( 0.05 ) ( 3 ) = P Simplify 1200 1.15 ≈ 1043.48 \begin{array}{l|rcl}
& A &=& P(1 + rt) \\
\text{Divide both sides by } (1 + rt) & \dfrac{A}{1 + rt} &=& P \\
\text{Plug in values} & \dfrac{1200}{1 + (0.05)(3)} &=& P \\
\text{Simplify} & \dfrac{1200}{1.15} &\approx& 1043.48
\end{array} Divide both sides by ( 1 + r t ) Plug in values Simplify A 1 + r t A 1 + ( 0.05 ) ( 3 ) 1200 1.15 1200 = = = ≈ P ( 1 + r t ) P P 1043.48
You need to invest about 1043.48 dollars.
Find the height - A triangle has area 30 and base 12. What’s the height?
Triangle — A = ½bh b h
A = 1 2 b h Multiply both sides by 2 2 A = b h Divide both sides by b 2 A b = h Plug in values 2 ( 30 ) 12 = 5 \begin{array}{l|rcl}
& A &=& \tfrac{1}{2}bh \\
\text{Multiply both sides by 2} & 2A &=& bh \\
\text{Divide both sides by b} & \dfrac{2A}{b} &=& h \\
\text{Plug in values} & \dfrac{2(30)}{12} &=& 5
\end{array} Multiply both sides by 2 Divide both sides by b Plug in values A 2 A b 2 A 12 2 ( 30 ) = = = = 2 1 bh bh h 5
The height is 5.
This skill lets you adapt formulas to whatever you’re solving for.
Literal equations are just regular solving with extra letters. Treat them like numbers. Move terms away from the variable you want, step by step. Practice with common formulas (distance, interest, area) and you’ll rearrange them quickly.
Solve for r in d = rt. A. r = d / t B. r = t / d C. r = d t D. r = d + t
Solve for h in A = (1/2)bh. A. h = 2A / b B. h = A / (2b) C. h = 2A b D. h = A b / 2
Solve for P in I = Prt (simple interest). A. P = I / (rt) B. P = I rt C. P = I + rt D. P = rt / I
Solve for m in y = mx + b. A. m = (y - b) / x B. m = y - b x C. m = (y + b) / x D. m = b / (y - x)
Solve for $b$ in $A = \frac{1}{2}(a + b)h$. A. $b = \frac{2A}{h} - a$ B. $b = 2Ah - a$ C. $b = \frac{A - a}{2h}$ D. $b = \frac{2A + a}{h}$
Retry Quiz
