Linear Sentences in One Variable
Rational Expressions
This article defines what proportion, direct variation, inverse variation, and joint variation are and explains how to solve such equations.
Proportion
A
proportion is an equation stating that two rational expressions are equal. Simple proportions can be solved by applying the cross products rule.
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More involved proportions are solved as rational equations.
Example 1: Solve
.
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Apply the cross products rule.
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The check is left to you.
Example 2: Solve
.
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Apply the cross products rule.
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The check is left to you.
Example 3: Solve
.
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However,
x = 4 is an extraneous solution, because it makes the LCD become zero. To check to see if
is a solution is left to you.
Direct variation
The phrase “ y varies directly as x” or “ y is directly proportional to x” can be translated in two ways.
-
for some constant
k.
The k is called the constant of proportionality. This translation is used when the constant is the desired result.
-
This translation is used when the desired result is either an original or new value of x or y.
Example 4: If
y varies directly as
x, and
y = 10 when
x = 7, find the constant of proportionality.
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The constant of proportionality is
.
Example 5: If
y varies directly as
x, and
y = 10 when
x = 7, find
y when
x = 12.
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Apply the cross products rule.
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Inverse variation
The phrase “ y varies inversely as x” or “ y is inversely proportional to x” is translated in two ways.
-
yx = k for some constant k, called the constant of proportionality. Use this translation if the constant is desired.
-
y1 x1 = y2 x2.
Use this translation if a value of x or y is desired.
Example 6: If
y varies inversely as
x, and
y = 4 when
x = 3, find the constant of proportionality.
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The constant is 12.
Example 7: If
y varies inversely as
x, and
y = 9 when
x = 2, find
y when
x = 3.
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Joint variation
If one variable varies as the product of other variables, it is called joint variation. The phrase “ y varies jointly as x and z” is translated in two ways.
-
if the constant is desired
-
if one of the variables is desired
Example 8: If
y varies jointly as
x and
z, and
y = 10 when
x = 4 and
x = 5, find the constant of proportionality.
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Example 9: If
y varies jointly as
x and
z, and
y = 12 when
x = 2 and
z = 3, find
y when
x = 7 and
z = 4.
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Occasionally, a problem involves both direct and inverse variations. Suppose that y varies directly as x and inversely as z. This involves three variables and can be translated in two ways:
-
if the constant is desired
-
Example 10: If
y varies directly as
x and inversely as
z, and
y = 5 when
x = 2 and
z = 4, find
y when
x = 3 and
z = 6.
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