Khan Academy Algebra Flashcards

Variables

• A variable is a symbol representing an unknown value.
• Examples of variables include letters like a and x in the expression a + 1 = x.

Algebraic Expressions and Equations

• An algebraic expression consists of terms but lacks an equal sign, such as 3x + 2.
• An equation represents a statement that two expressions are equal, demonstrated as 2x + 1 = 5.

Dependent and Independent Variables

• A dependent variable is measured and typically plotted on the y-axis.
• An independent variable is manipulated and is usually placed on the x-axis.
• For instance, in the scenario where John works x hours at $3 per hour, his total earnings y are dependent on x, resulting in the equation 3x = y.

Combining Like Terms

• Terms with the same variable can be combined to simplify expressions, e.g., 2x + 3y - 2x simplifies to 1x + 3y.
• Proper distribution of parentheses is required before combining, such as turning 2x(3x + 25) into 6x² + 48x.

Equivalent Expressions

• Equivalent expressions represent the same value regardless of their form, e.g., 4a + 8 is equivalent to 2(2a + 4).

Types of Equations

• A linear equation has its highest power as 1, exemplified by 2x + 2.
• Dividing by zero results in an undefined scenario within mathematics.

Solutions to Equations

• An equation has one solution when a variable equals a specific number, such as x = 5.
• An equation has zero solutions when conflicting values exist on both sides, like 4 = 6.
• An equation has infinite solutions when both sides are identically equal, e.g., 13 = 13.

Integer Relationships

• The sum of consecutive odd or even integers can be expressed with x, x + 2, x + 4 to capture the relationship between integers.

Unit Rates

• A unit rate is defined as a rate whose denominator is one, essential for comparing quantities.
• Setting up a unit rate involves dividing the dependent variable by the independent variable, such as distance per hour based on a speed, exemplified by 45 m/hr.

Inequalities

• When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be flipped. For example, -2x > 4 would change the inequality if isolating x.