



Manipulating Mathematical Expressions
or
How to solve for x
 RULE
 Any operation my be performed on an equation, provided the same
operation is applied to both sides of the equals sign.
 EXAMPLE
 Given the equation:
2x + 4y = 12
Through mathematical manipulations, this
expression can be transformed into one which defines x in terms of y.
 Start with the original formula
2x + 4y = 12
 Examine the formula. What changes need to be made to obtain an
expression for x in terms of y? In this case:
 Subtract 4y from both sides:
2x + 4y  4y = 12  4y
2x = 12  4y
 Divide both sides by 2:
2x/2 = (12  4y)/2

Determine the final formula:
x = 6  2y
If the value of y is known, the value of x may be determined using
this formula. For example, if y is 2, the value of x is 2.
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Chemistry problems are often solved using formulas. Generally, the
formula is manipulated to give the appropriate expression for the
problem in question and then the given values are substituted into the
formula. The area of a rectangle is found by multiplying the lengths
of the base and height together. The formula for the area of a
rectangle is:
A=bh
If the base of a rectangle is 10 cm and the height is 4 cm the area is
given by:
A = 10 cm x 4 cm
A = 40 cm
Note: the units multiply along with the numbers. cm x cm =
cm^{2}
Manipulating formulasI
If the area and the base of a rectangle are given, the height can be
found by manipulating the original formula:
 Start with the original formula:
A=bh
 Examine the formula. What changes need to be made to obtain an
expression for the length of the height (h) in terms of the other
variables (Area (A) and length of base (b))?
Divide both sides by the length of the base b:
A/b = (bh)/b
 Determine the final formula:
A/b = h
If the area of a rectangle is 36 cm^{2} and the base is 6 cm,
the height can be found using this new formula:
h = A/b = 36cm^{2}/6cm = 6 cm
Manipulating formulasII
Some formulas require more manipulation but the basic procedure is the
same. Consider the formula for the area of a circle:
Area = x (radius)^{2}
If the area of a circle is known, the radius can be found using the
forumla for area:
 Start with the original formula:
A = r^{ 2}
 Examine the formula. What changes need to be made to obtain an
expression for the radius (r) in terms of the other variable (Area,
(A))? In this case, a constant is
also involved.
 Divide both sides by :
A/ = (r ^{2})/
A/ =r ^{2}
 Take the square root of both sides:
√ (A/) = √ (r
^{2})
 Determine the final formula:
r = √ (A/)
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