Algebra Essentials
87 flashcards by MentisLibrary
The rules that carry you from combining like terms to logarithms, each one paired with a worked example and the mistake that trips people up. Build the fluency that later math courses quietly assume.
- What is the order of operations?
- Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
- What is the distributive property?
- $a(b + c) = ab + ac$
- What is the commutative property?
- Order does not matter for addition or multiplication: $a + b = b + a$ and $ab = ba$
- What is the associative property?
- Grouping does not matter for addition or multiplication: $(a+b)+c = a+(b+c)$
- What are like terms?
- Terms with exactly the same variables raised to exactly the same powers, so they can be added or subtracted
- What are the coefficient, variable, and constant in $7x + 4$?
- $7$ is the coefficient, $x$ is the variable, $4$ is the constant
- How do you solve a linear equation in one variable?
- Undo the operations in reverse order, doing the same thing to both sides, until the variable is alone
- How do you solve an equation with the variable on both sides?
- Collect variable terms on one side and constants on the other, then divide by the coefficient
- What does $|x|$ mean?
- The distance of $x$ from zero on the number line, so it is never negative
- How do you solve $|ax + b| = c$ when $c > 0$?
- Split into two equations: $ax + b = c$ and $ax + b = -c$
- How do you solve $|x| < c$ versus $|x| > c$?
- $|x| < c$ becomes $-c < x < c$ (an and). $|x| > c$ becomes $x < -c$ or $x > c$ (an or)
- When does an inequality sign flip?
- When you multiply or divide both sides by a negative number
- How do you write the solution set $-3 < x \le 5$ in interval notation?
- $(-3, 5]$
- What is the slope formula?
- $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
- What is slope-intercept form?
- $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept
- What is point-slope form?
- $y - y_1 = m(x - x_1)$ for a line through $(x_1, y_1)$ with slope $m$
- What is the standard form of a line?
- $Ax + By = C$, with $A$, $B$, and $C$ usually integers
- How are the slopes of parallel lines related?
- They are equal
- How are the slopes of perpendicular lines related?
- They are negative reciprocals, so $m_1 m_2 = -1$
- How do you find the $x$- and $y$-intercepts?
- Set $y = 0$ and solve for $x$, then set $x = 0$ and solve for $y$
- How do you solve a system by substitution?
- Solve one equation for one variable, substitute into the other, then back-substitute
- How do you solve a system by elimination?
- Scale one or both equations so a variable cancels when you add them
- What do no solution and infinitely many solutions look like in a system?
- No solution means parallel lines and a false statement like $0 = 5$. Infinitely many means the same line and a true statement like $0 = 0$
- What is the product rule for exponents?
- $a^m \cdot a^n = a^{m+n}$
- What is the quotient rule for exponents?
- $\dfrac{a^m}{a^n} = a^{m-n}$
- What is the power of a power rule?
- $(a^m)^n = a^{mn}$
- What is the power of a product rule?
- $(ab)^n = a^n b^n$
- What is $a^0$ for $a \neq 0$?
- $a^0 = 1$
- What does a negative exponent mean?
- $a^{-n} = \dfrac{1}{a^n}$
- What does a fractional exponent mean?
- $a^{m/n} = \sqrt[n]{a^m}$, so the denominator is the root and the numerator is the power