A literal equation is, simply put, an equation that has a lot of letters or
variables.
For example,
A = π β w
(The formula for finding the area of a rectangle)
E = mc2
(Einsteinβs Theory of Relativity)
are both literal equations.
When a literal equation is given, we would often be asked to solve the equation for a
given variable. The goal is to isolate that given variable. The process is the same
process that we use to solve linear equations; the only difference is that we will be working with a lot more letters, and we may not be able to simplify as much as we can with linear equations.
Example 1 :
Solve for w in the formula for area of a rectangle :
A = π β w
Solution :
A = π β w
Divide each side by π.
ᴬβπ = w
Example 2 :
Solve for c in the formula for Einsteinβs Theory of Relativity :
E = mc2
Solution :
E = mc2
Divide each side by m.
α΄±βm = c2
Take square root on each side.
β(α΄±βm) = βc2
β(α΄±βm) = c
Example 3 :
Solve for h in the formula for the surface area of a right cylinder :
S = 2Οr(h + r)
Solution :
S = 2Οr(h + r)
Use distributive property of multiplication over addition.
S = 2Οrh + 2Οr2
Subtract 2Οr2 from each side.
S - 2Οr2 = 2Οrh
Divide each side by 2Οr.
Example 4 :
Solve for r in the formula for volume of sphere :
V = β΄ββ β Οr3
Solution :
V = β΄ββ β Οr3
Multiply each side by 3/4.
Β³β±½ββ = Οr3
Divide each side by Ο.
Β³β±½ββΟ = r3
Take cube root on each side.
3β(Β³β±½ββΟ) = 3βr3
3β(Β³β±½ββΟ) = r
Example 5 :
Solve for w in the formula for perimeter of the rectangle :
P = 2(π + w)
Solution :
P = 2(π + w)
Use distributive property of multiplication over addition.
P = 2π + 2w
Subtract 2π from each side.
P - 2π = 2w
Divide each side by 2.
β½α΄Ύ β» Β²πβΎββ = w
Example 6 :
Solve for a :
Q = 3a + 5ac
Solution :
Q = 3a + 5ac
Factor a out from (3a + 5ac).
Q = a(3 + 5c)
Divide each side by (3 + 5c).
Qβββ β β cβ = a
Example 7 :
Solve for b1 :
A = Β½ β
h(b1 + b2)
Solution :
A = Β½ β h(b1 + b2)
A = Κ°ββ β (b1 + b2)
Multiply each side by Β²βh.
²ᴬβh = b1 + b2
Subtract b2 from each side.
²ᴬβh - b2 = b1
Example 8 :
Solve for b :
a2 + b2 = c2
Solution :
a2 + b2 = c2
Subtract a2 from each side.
b2 = c2 - a2
Take square root on each side.
βb2 = β(c2 - a2)
b = β(c2 - a2)
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