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11.7 Summary of key concepts

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module reviews the key concepts from the chapter "Algebraic Expressions and Equations."

Summary of key concepts

Numerical expression ( [link] )

A numerical expression results when numbers are associated by arithmetic opera­tion signs. The expressions 3 + 5 size 12{3+5} {} , 9 − 2 size 12{9 - 2} {} , 5 ⋅ 6 size 12{5 cdot 6} {} and 8 ÷ 5 size 12{8 div 5} {} are numerical expressions.

Algebraic expressions ( [link] )

When an arithmetic operation sign connects a letter with a number or a letter with a letter, an algebraic expression results. The expressions 4 x + 1 size 12{4x+1} {} , x − 5 size 12{x - 5} {} , 7 x ⋅ 6 y size 12{7x cdot 6y} {} , and 4 x ÷ 3 size 12{4x div 3} {} are algebraic expressions.

Terms and factors ( [link] )

Terms are parts of sums and are therefore separated by addition (or subtraction) signs. In the expression, 5 x − 2 y size 12{5x - 2y} {} , 5 x size 12{5x} {} and − 2 y size 12{ - 2y} {} are the terms. Factors are parts of products and are therefore separated by multiplication signs. In the expression 5 a , 5 and a are the factors.

Coefficients ( [link] )

The coefficient of a quantity records how many of that quantity there are. In the expression 7 x size 12{7x} {} , the coefficient 7 indicates that there are seven x size 12{x} {} 's.

Numerical evaluation ( [link] )

Numerical evaluation is the process of determining the value of an algebraic ex­pression by replacing the variables in the expression with specified values.

Combining like terms ( [link] )

An algebraic expression may be simplified by combining like terms. To combine like terms , we simply add or subtract their coefficients then affix the variable. For example 4 x + 9 x = ( 4 + 9 ) x = 13 x size 12{4x+9x= \( 4+9 \) x="13"x} {} .

Equation ( [link] )

An equation is a statement that two expressions are equal. The statements 5 x + 1 = 3 size 12{5x+1=3} {} and 4 x 5 + 4 = 2 5 size 12{ { {4x} over {5} } +4= { {2} over {5} } } {} are equations. The expressions represent the same quantities.

Conditional equation ( [link] )

A conditional equation is an equation whose truth depends on the value selected for the variable. The equation 3 x = 9 size 12{3x=9} {} is a conditional equation since it is only true on the condition that 3 is selected for x size 12{x} {} .

Solutions and solving an equation ( [link] )

The values that when substituted for the variables make the equation true are called the solutions of the equation.
An equation has been solved when all its solutions have been found.

Equivalent equations ( [link] )

Equations that have precisely the same solutions are called equivalent equations . The equations 6 y = 18 size 12{6y="18"} {} and y = 3 size 12{y=3} {} are equivalent equations.

Addition/subtraction property of equality ( [link] )

Given any equation, we can obtain an equivalent equation by
  1. adding the same number to both sides, or
  2. subtracting the same number from both sides.

Solving x + a = b And x - a = b ( [link] )

To solve x + a = b size 12{x+a=b} {} , subtract a size 12{a} {} from both sides.
x + a = b x + a − a = b − a x = b − a
To solve x − a = b size 12{x - a=b} {} , add a to both sides.
x − a = b x − a + a = b + a x = b + a

Multiplication/division property of equality ( [link] )

Given any equation, we can obtain an equivalent equation by
  1. multiplying both sides by the same nonzero number, that is, if c ≠ 0 size 12{c<>0} {} , a = b size 12{a=b} {} and a ⋅ c = b ⋅ c size 12{a cdot c=b cdot c} {} are equivalent.
  2. dividing both sides by the same nonzero number, that is, if c ≠ 0 size 12{c<>0} {} , a = b size 12{a=b} {} and a c = b c size 12{ { {a} over {c} } = { {b} over {c} } } {} are equivalent.

Solving a x = b And x a = b ( [link] )

To solve ax = b size 12{ ital "ax"=b} {} , a ≠ 0 size 12{a<>0} {} , divide both sides by a size 12{a} {} .
a x = b a x a = b a a x a = b a x = b a
To solve x a = b size 12{ { {x} over {a} } =b} {} , a ≠ 0 size 12{a<>0} {} , multiply both sides by a size 12{a} {} .
x a = b a ⋅ x a = a ⋅ b a ⋅ x a = a ⋅ b x = a ⋅ b

Translating words to mathematics ( [link] )

In solving applied problems, it is important to be able to translate phrases and sentences to mathematical expressions and equations.

The five-step method for solving applied problems ( [link] )

To solve problems algebraically, it is a good idea to use the following five-step procedure .
After working your way through the problem carefully, phrase by phrase:
  1. Let x size 12{x} {} (or some other letter) represent the unknown quantity.
  2. Translate the phrases and sentences to mathematical symbols and form an equation. Draw a picture if possible.
  3. Solve this equation.
  4. Check the solution by substituting the result into the original statement of the problem.
  5. Write a conclusion.
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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