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 = mc^{2}
(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.
α΄¬βπ = w
Example 2 :
Solve for c in the formula for Einsteinβs Theory of Relativity :
E = mc^{2}
Solution :
E = mc^{2}
Divide each side by m.
α΄±βm = c^{2}
Take square root on each side.
β(α΄±βm) = βc^{2}
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 + 2Οr^{2}
Subtract 2Οr^{2 }from each side.
S - 2Οr^{2 }= 2Οrh
Divide each side by 2Οr.
Example 4 :
Solve for r in the formula for volume of sphere :
V = β΄ββ β Οr^{3}
Solution :
V = β΄ββ β Οr^{3}
Multiply each side by 3/4.
Β³β±½ββ = Οr^{3}
Divide each side by Ο.
Β³β±½ββΟ = r^{3}
Take cube root on each side.
^{3}β(Β³β±½ββΟ) = ^{3}βr^{3}
^{3}β(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» Β²^{π}βΎββ = 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 b_{1} :
A = Β½ β
h(b_{1} + b_{2})
Solution :
A = Β½ β h(b_{1} + b_{2})
A = Κ°ββ β (b_{1} + b_{2})
Multiply each side by Β²βh.
Β²α΄¬βh = b_{1} + b_{2}
Subtract b_{2} from each side.
Β²α΄¬βh - b_{2 }= b_{1}
Example 8 :
Solve for b :
a^{2} + b^{2} = c^{2}
Solution :
a^{2} + b^{2} = c^{2}
Subtract a^{2} from each side.
b^{2} = c^{2 }- a^{2}
Take square root on each side.
βb^{2} = β(c^{2 }- a^{2})
b = β(c^{2 }- a^{2})
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