(1) What are variables, expressions, and equations? | Introduction to algebra | Algebra I | Khan Academy

Basic Arithmetic

• Basic arithmetic involves concrete numbers and straightforward calculations.

  • Example: 23 plus 5 equals 28.

  • Other operations: 2 times 7, 3 divided by 4, etc.

Introduction to Algebra

• Transition from basic arithmetic to algebra introduces variables.

• Variables: symbols that represent values and can change.

  • Example: In the expression x + 5, the value changes based on x.

Evaluating Expressions

• When x = 1:

  • Expression: x + 5 = 1 + 5 = 6.

• When x = -7:

  • Expression: x + 5 = -7 + 5 = -2.

• Variables can take different values based on context.

Expressions vs. Equations

• Expression: A statement of value or quantity that changes with variable values.

  • Example: x + 5, y + z (evaluated based on variable values).

• Equation: A statement that two expressions are equal.

  • Example: x + 3 = 1.

  • This can be solved to find the value of x (x = -2 in this case).

Constraints in Equations

• Equations can constrain variable values based on the relationships established.

  • Example: x + y + z = 5.

  • If y = 3 and z = 2, then:

    • x + 5 = 5.

    • Therefore, x = 0.

Importance of Context in Variables

• A variable's value can vary based on problem context.

• Evaluating expressions with different variable values:

  • Example: Expression x^y. If x = 5 and y = 2:

    • Calculation: 5^2 = 25.

  • If x = -2 and y = 3:

    • Calculation: (-2)^3 = -8.

Complex Expressions

• More intricate expressions can be evaluated step by step:

  • Example:

    • Expression: √(x + y) - x. If x = 1 and y = 8:

    • Calculation:

      • Substitute: √(1 + 8) - 1.

      • Evaluate: √9 - 1 = 3 - 1 = 2.