1.16: Algebra Review Guide

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Kendra Meade and Anthony Assibi Mahama

Linear Equations, Formulas, and Inequalities

Eliminate denominators (multiply by Least Common Denominator)
Remove parentheses (distribute)
Get variable terms on one side (add/subtract principles)
Combine like terms (for formulas, factor the variable if it appears in more than 1 term)
Get the variable alone (multiply/divide principles)
Note: For inequalities, division or multiplication by a negative number will switch the inequality symbol around
Check the solution

Practice Problems

An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://iastate.pressbooks.pub/quantitativeplantbreeding/?p=339#h5p-79

An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://iastate.pressbooks.pub/quantitativeplantbreeding/?p=339#h5p-80

An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://iastate.pressbooks.pub/quantitativeplantbreeding/?p=339#h5p-81

An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://iastate.pressbooks.pub/quantitativeplantbreeding/?p=339#h5p-82

An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://iastate.pressbooks.pub/quantitativeplantbreeding/?p=339#h5p-83

An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://iastate.pressbooks.pub/quantitativeplantbreeding/?p=339#h5p-84

An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://iastate.pressbooks.pub/quantitativeplantbreeding/?p=339#h5p-85

Terminology

Various terms are used in carrying out algebraic operations. A good understanding of these enables calculations to be carried out correctly. Some common terms are shown in Table 1.

Table 1 Algebraic vocabulary terms and their definition.
Vocabulary	Definition
Variable	A letter that can be replaced by any number
Constant	A capital letter that represents a fixed number
Expression	Consists of variables, numbers and operation symbols (ex: $\small{2x - 5 + 3y^{2} - \frac{x}{3}}$ )
Equation	When “=” is between two algebraic expressions (equation can be true $\small{x + 2 = x + 2}$ , false $\small{x+2 = x + 3}$ or neither $\small{\textrm{or 3x -9 = 4x - 14}}$ )
Solution	A value replacing the variable to make the equation true ( $\small{x = 5}$ for $\small{3x - 9 = 4x - 14}$ )
Solved	Having found the values that make the equation true
Translate	To convert words to an algebraic expression or equation
Substitute	Replacing a variable with a given number
Evaluate	To find a solution
Factor	Multiple or Product
Factoring	Reversing distributive law and turning it into the factors (multiples)
Equivalent	Expressions that, when evaluated, produce the same value
Terms	A number, var (variable), or a product/quotient of numbers and/or variables separated by + or – signs (Expression $\small{5 - 3xy - \normalsize \frac{2x}{5}}$ contains three terms)
Fraction notation	A way of showing the division of two numbers (Numerator: top, Denominator: bottom)
Undefined	$\small{ \frac{a}{0}}$ : Division by zero is undefined — There is no solution
Zero fraction	$\small{ \frac{0}{a}}$ : Zero in the numerator makes fraction equal to zero
Fraction notation for 1	Any nonzero number divided by itself is 1: $\small{ \frac{a}{a} = 1}$
Reciprocal	Multiplicative inverse: if $\small{ a\neq 0}$ , $\small{ \frac{a\cdot 1}{a}=\frac{a}{b}\cdot \frac{b}{a} =1}$ Zero has no reciprocals. The reciprocal of a is $\small{a=\frac{1}{a}}$ .
Opposite	Additive inverse: opposite of “a” is “-a” and $\small{a+(-a)=0}$ .
Prime number	A natural number that is divisible by only two different factors: itself and 1
Inequality	A statement about the relative size of two objects. Indicated by <, >, ≪, ≫
Law	Definition
Commutative	For any real a and b, $\small{a+b=b+a}$ , and $\small{ a \cdot b = b \cdot a}$
Associative	For any real a, b, and c, $\small{a\cdot (b+c)=a \cdot b + a \cdot c}$
Distributive	For any real a, b and c, $\small{ \textrm{a + (b + c)}=(a + b) + c, {a\cdot (b\cdot c)=(a\cdot b)\cdot c}}$

Operation With Fractions

Simplifying Fractions

Prime factorize each numerator and denominator
Remove factors that are the same from the numerator and denominator and replace them with “1”.
These form fractions that are equal to “1”.
Multiply the remaining factors in the numerator
Multiply the remaining factors in the denominator

Simplifying Fractions Example

\[\frac{60}{126}=2^{2}\cdot 3\cdot \frac{5}{2}\cdot 3^{2}\cdot 7=\frac{10}{21}\]

Multiplying Fractions

Prime factorize each numerator and denominator
Remove factors that are the same from any numerator and any denominator and replace them with “1”. These form fractions that are equal to “1”.
Multiply the remaining factors in all numerators
Multiply the remaining factors in all denominators

Multiplying Fractions Example

\[\frac{7}{12}\cdot \frac{30}{21}=\frac{7}{2^{2}}\cdot 3\cdot 2\cdot 3\cdot \frac{5}{3}=\frac{5}{6}\]

Dividing Fractions

Change division to multiplication by the inverse of the second fraction
Multiply as above

Dividing Fractions Example

\[\frac{4}{18}\div\frac{10}{21}=\frac{4}{18}\cdot {\frac{21}{10}}=\frac{2^{2}}{2}\cdot 3^{2}\cdot 3\cdot \frac{7}{2}\cdot 5=\frac{1}{3}\cdot \frac{7}{5}=\frac{7}{15}\]

Add and Subtract

If denominators are the same, go to step 4
Otherwise, prime factorize each denominator
Find the Least Common Denominator (LCD) by multiplying each fraction by a fraction equal to “1”, made out of the missing factors from each denominator
Once denominators are the same, add/subtract numerators and write over the LCD
Simplify as above

Adding/Subtracting Fractions Example 1: Same denominator

\[\frac{2}{14}+\frac{5}{14}=\frac{7}{14}=\frac{7}{2}\cdot 7=\frac{1}{2}\]

Adding/Subtracting Fractions Example 2: One denominator is a multiple of the other

\[\frac{3}{2}-\frac{5}{6}=\frac{3}{2}-\frac{5}{2}\cdot 3=\frac{3}{2}\cdot \frac{3}{3}-\frac{5}{2}\cdot 3=9-\frac{5}{2}\cdot 3=\frac{4}{2}\cdot 3=\frac{2^{2}}{2}\cdot 3=\frac{2}{3}\]

Adding/Subtracting Fractions Example 3: Different denominators

\[\frac{9}{42}-\frac{3}{15}=\frac{9}{2}\cdot 3\cdot 7-\frac{3}{3}\cdot 5=\frac{9}{2}\cdot 3\cdot 7\cdot \frac{5}{5}-\frac{3}{3}\cdot 5\cdot 2\cdot \frac{7}{2}\cdot 7=45-\frac{42}{2}\cdot 3\cdot 5\cdot 7=32\cdot 3\cdot 5\cdot 7=\frac{1}{2}\cdot 5\cdot 7=\frac{1}{70}\]

Rules of Exponents

Zero and One

$a^{0}=1$
$a^{1}=a$

Multiply and Divide

$\frac{a}{b}\times\frac{a}{n}=a^{b+n}$
$\frac{a^{b}}{a^{n}}=a^{b-n}$

Distribute

$(a^{b}c^{n})^p=a^{bp}c^{np}$
$(\frac{a^{b}}{c^{n}})^p=a^{bp}c^{np}$

Negative

$a^{-n}=\frac{1}{a^{n}}$
$\frac{1}{a^{-n}}=a^{n}$
$\frac{a^{-n}}{b^{-c}}=\frac{b^{c}}{a^{n}}$

Rules of Radicals

Add and Subtract Radicals

Simplify each radical by removing perfect square roots out of the radical to get like radicals, then combine the number of like radicals.

Multiply and Divide Radicals

\[\sqrt[n]{a}\cdot\sqrt[n]{b} =\sqrt[n]{ab}\]

\[\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\]

If no n is given, assume that n is 2, and a square root is required.

Plus/Minus Sign

Used to indicate that a value can be of either sign. Often used to construct confidence intervals.

\[p\pm x + y = p +x + y, \textrm{or p - x + y}\]

Data Transformation

Trigonometry and Natural Logarithms

Table 2 Trigonometric transformation terms and their reciprocals.
Transformation	Reciprocal of Transformation
Sine (sin)	Cosecant (csc or cosec) or Arcsine (arcsin)
Cosine (cos)	Secant (sec) or Arccosine (arccos)
Tangent (tan)	Cotangent (cot) or Arctangent (arctan)
Arcsine (arcsin)	Sine (sin)
Natural logarithm (in or log or log_e)	Exponential (exp or eⁿ)

Summation

Summation (S) signifies that a series of terms should be added together.

Given a series of numbers $x_1$ , $x_2$ , $x_3$ … $x_n$

Sum all $x_i$ starting at $i=1$ through $i=n$ .

\[\sum_{i=1}^{n}x^{i}=x_{1}+x_{2}+x_{3}+x_{4} + ... + x_{n}\]

\[\textrm{Equation 1}\]

How to cite this chapter: Kendra Meade and A. A. Mahama. 2023. Algebra Review Guide. In W. P. Suza, & K. R. Lamkey (Eds.), Quantitative Methods. Iowa State University Digital Press.

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