Algebra Chapter: Variables and Expressions

• Using letters to represent numbers in mathematics, also known as variables, is a fundamental concept. It allows for generalizing mathematical relationships and solving problems more efficiently.

Variables and Expressions

• Variables are symbols, usually letters (like x, y, or z), that stand for unknown or unspecified numerical values.
• Replacing a numerical value with a variable creates an algebraic expression. For example, "2x + 5" is an algebraic expression.
• The numerical values assigned to the variables are called solutions or roots of an equation.
• Evaluating an expression means substituting the variable with a given number and performing the corresponding operations. For instance, if x = 3 in the expression 2x + 5, the result is 2(3) + 5 = 11.

Equations and Inequalities

• An equation states that two expressions are equal. A simple example is 2x + 3 = 7.
• Solving an equation involves finding the value(s) of the variable(s) that make the equation true. In the example above, x = 2.
• An inequality compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), or ≠ (not equal to). For example, y ≤ 10 represents that y can take on any value up to, and including, 10.

Properties of Equality

• The properties of equality are essential for solving equations. Key ones include:

  • Addition Property of Equality: If a = b, then a + c = b + c
  • Subtraction Property of Equality: If a = b, then a - c = b - c
  • Multiplication Property of Equality: If a = b, then a * c = b * c
  • Division Property of Equality: If a = b, and c ≠ 0, then a / c = b / c

• These properties allow transforming equations into equivalent forms that are easier to solve.

Applications of Variables

• Variables are used extensively in various mathematical situations, allowing creation and investigation of widespread algebraic patterns.
  • Representing unknown quantities in word problems.
  • Defining functions and graphing relationships between variables.
  • Modeling real-world phenomena using mathematical equations.
  • Creating and manipulating formulas, often containing multiple variables.

Simplifying Expressions

• Simplifying algebraic expressions means rewriting them in their most basic or standard form. This involves applying the order of operations (PEMDAS/BODMAS) and combining like terms.
• For example, the expression 3x + 2x simplifies to 5x.

Solving Linear Equations

• Linear equations are equations where the highest power of the variable is 1 (e.g., 2x + 5 = 11).

• The techniques for solving linear equations often involve using the properties of equality to isolate the variable on one side of the equation.

• Techniques typically involve:

  • Combining like terms, and then
  • Using the addition or subtraction property of equality to move constant terms to one side of the equation
  • Using multiplication or division to isolate the variable, if necessary.

Word Problems

• Applying variables to solve word problems involves first defining the variable that represents an unknown quantity. Next, setting up an equation or system of equations that models the relationships described in the problem. Finally, solve for the unknown using the mathematical tools learned.

• For instance, "If John has 5 more apples than Mary, and together they have 17 apples, how many apples does Mary have?" "x" could represent the number of apples Mary has, then "x + 5" equals the number of apples John has, and "x + (x + 5) = 17."