Solving Equations
Chapter 2, Section 1
- Properties of Equality
- For all real numbers a, b, c:
-
| Reflexive Property
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a = a
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| Symmetric Property
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If a = b, then b = a
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| Transitive Property
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If a = b and b = c, then a = c
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| Addition Property of Equality
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If a = b, then a + c = b + c
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| Multiplication Property of Equality
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If a = b, then a x c = b x c
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Proportions
(cross multiply)
|
| If
|
a
b
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=
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c
d
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with b, d  0,
|
| then ad = bc.
|
|
- Combining Terms
- Defn: The coefficient is the numerical part of a term that
precedes the variable.
- Defn: The degree of a term is the sum of the exponents on the variables
in the term.
- Defn: Like terms have the same variables with the same
exponents.
- To simplify an expression means to combine all like terms in the expression.
- Equations
- Defn: A solution (or root) of an equation is a number that
makes the equation
true when that number is substituted in for the variable.
- Equations may have one solution, no solution, or many solutions:
| Conditional Equation
|
Has exactly one real solution.
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| Identity
|
Is true for all real numbers--has an infinite
number of solutions.
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| Inconsistent Equation
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Has no solution
|
- Solving Linear Equations in One Variable
- Defn: A linear equation in one variable is a first-degree equation
(largest exponent on the variable is 1) with only one variable.
- A linear equation in one variable may always be written in the form ax = b.
- Trick to solving: Use the properties of equality to
get the given equation into an equivalent equation of the
form ax = b. Then the solution is
| Steps to Solving a Linear Equation
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| Eliminate fractions by multiplying both sides by the
least common denominator.
|
| Remove grouping symbols
(as in "order of operations," Chapter1, section4).
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| Combine like terms on each side of the equal sign.
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| Use addition property of equality (maybe repeatedly) to
get the equation into the form ax = b.
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| Divide both sides by a.
The solution is
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| Check your solution in the original
equation by substitution.
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