Linear Equations, Formulas, and Inequalities

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

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.

Operation With Fractions

Simplifying Fractions

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

Multiplying Fractions

1. Prime factorize each numerator and denominator
2. 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”.
3. Multiply the remaining factors in all numerators
4. Multiply the remaining factors in all denominators

Multiplying Fractions Example

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

Dividing Fractions

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

Dividing Fractions Example

[latex]\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}[/latex]

Add and Subtract

1. If denominators are the same, go to step 4
2. Otherwise, prime factorize each denominator
3. 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
4. Once denominators are the same, add/subtract numerators and write over the LCD
5. Simplify as above

Rules of Exponents

Zero and One

• [latex]a^{0}=1[/latex]
• [latex]a^{1}=a[/latex]

Multiply and Divide

• [latex]\frac{a}{b}\times\frac{a}{n}=a^{b+n}[/latex]
• [latex]\frac{a^{b}}{a^{n}}=a^{b-n}[/latex]

Distribute

• [latex](a^{b}c^{n})^p=a^{bp}c^{np}[/latex]
• [latex](\frac{a^{b}}{c^{n}})^p=a^{bp}c^{np}[/latex]

Negative

• [latex]a^{-n}=\frac{1}{a^{n}}[/latex]
• [latex]\frac{1}{a^{-n}}=a^{n}[/latex]
• [latex]\frac{a^{-n}}{b^{-c}}=\frac{b^{c}}{a^{n}}[/latex]

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

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

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

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.

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

Data Transformation

Trigonometry and Natural Logarithms

Summation

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

Given a series of numbers [latex]x_1[/latex], [latex]x_2[/latex], [latex]x_3[/latex] … [latex]x_n[/latex]

Sum all [latex]x_i[/latex] starting at [latex]i=1[/latex] through [latex]i=n[/latex].

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

[latex]\textrm{Equation 1}[/latex] Equation for summing series of values.