11.7: Summary of Key Concepts

Summary of Key Concepts

Numerical Expression
A numerical expression results when numbers are associated by arithmetic opera­tion signs. The expressions 3 + 5, 9 - 2, \(5 \cdot 6\) and \(8 \div 5\) are numerical expressions.

Algebraic Expressions
When an arithmetic operation sign connects a letter with a number or a letter with a letter, an algebraic expression results. The expressions \(4x + 1, x - 5, 7x \cdot 6y\), and \(4x \div 3\) are algebraic expressions.

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

Coefficients
The coefficient of a quantity records how many of that quantity there are. In the expression \(7x\), the coefficient 7 indicates that there are seven \(x\)'s.

Numerical Evaluation
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
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 \(4x + 9x = (4 + 9) x = 13x\).

Equation
An equation is a statement that two expressions are equal. The statements \(5x + 1 = 3\) and \(\dfrac{4x}{5} + 4 = \dfrac{2}{5}\) are equations. The expressions represent the same quantities.

Conditional Equation
A conditional equation is an equation whose truth depends on the value selected for the variable. The equation \(3x = 9\) is a conditional equation since it is only true on the condition that 3 is selected for \(x\).

Solutions and Solving an Equation
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
Equations that have precisely the same solutions are called equivalent equations. The equations \(6y = 18\) and \(y = 3\) are equivalent equations.

Addition/Subtraction Property of Equality
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 \(x - a = b\)
To solve \(x + a = b\), subtract \(a\) from both sides.

\(x + a = b\)
\(x + a - b = b - a\)
\(x = b - a\)

To solve \(x - a = b\), add \(a\) to both sides.

\(x - a = b\)
\(x - a + b = b + a\)
\(x = b + a\)

Multiplication/Division Property of Equality
Given any equation, we can obtain an equivalent equation by

1. multiplying both sides by the same nonzero number, that is, if \(c ne 0\), \(a = b\) and \(a \cdot c = b \cdot c\) are equivalent.
2. dividing both sides by the same nonzero number, that is, if \(c \ne 0\), \(a = b\) and \(\dfrac{a}{c} = \dfrac{b}{c}\) are equivalent.

Solving \(ax = b\) and \(\dfrac{x}{a} = b\)
To solve \(ax = b\), \(a \ne 0\), divide both sides by \(a\).

\(\begin{array} {rcl} {ax} & = & {b} \\ {\dfrac{ax}{a}} & = & {\dfrac{b}{a}} \\ {\dfrac{\cancel{a} x}{\cancel{a}}} & = & {\dfrac{b}{a}} \\ {x} & = & {\dfrac{b}{a}} \end{array}\)

To solve \(\dfrac{x}{a} = b, a \ne 0\), multiply both sdies by \(a\).

\(\begin{array} {rcl} {\dfrac{x}{a}} & = & {b} \\ {a \cdot \dfrac{x}{a}} & = & {a \cdot b} \\ {\cancel{a} \cdot \dfrac{x}{\cancel{a}}} & = & {a \cdot b} \\ {x} & = & {a \cdot b} \end{array}\)

Translating Words to Mathematics
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
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 \(x\) (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.