Variables and Expressions (Algebra I)

Variables and Expressions

• Variable: a symbol, generally a lowercase letter, used to represent an unknown quantity. It acts as a placeholder.

• Commonly used letters: x, y, and z. You can use any lowercase letter, but x, y, z are the ones you'll see most often early in algebra.

• Purpose: to model quantities that can vary or be unknown in real-world situations.

• An expression is a combination of numbers, variables, and operations that represents a value (e.g., $63 + x$).

• In this lesson, we contrast a known amount with an unknown amount: a constant like 63 dollars per day versus a variable representing tips.

Using Variables to Model Real-World Scenarios

• Example 1: Mark’s daily earnings at a car wash.

  • Known base: $63$ dollars per day (the minimum he earns).

  • Unknown addition: tips, represented by the variable $x$.

  • Model (expression): earnings = $63 + x$.

  • Interpretation: x is the amount of tips earned that day.

  • Substitution practice:

  • If the day’s tips are $7,$ then set $x = 7$ and compute $63 + 7 = 70.$ So he earns $70$ that day.

  • If the day’s tips are $9,$ then set $x = 9$ and compute $63 + 9 = 72.$ So he earns $72$ that day.

  • Point: as the value of $x$ changes, the total earnings change accordingly, illustrating how the same expression yields different results.

• Example 2: Jason buys two gallons of milk at an unknown price per gallon.

  • Known quantity: two gallons.

  • Unknown price: per-gallon cost is represented by the variable $y$.

  • Model (expression): total cost = number of gallons × price per gallon = $2 imes y = 2y$.

  • Notation note: when a number is next to a variable, multiplication is implied. So $2y$ means two times the price per gallon.

  • Rationale for notation: using the juxtaposition (2y) avoids confusion with the variable x (which is also common in algebra).

  • Alternative valid notations (all mean the same): $2y$, $2(y)$, or $(2)y$.

  • Substitution example: if the price per gallon is $y = 1$ (no dollar sign needed in the algebraic expression), then the total cost for two gallons is $2 imes 1 = 2$ dollars; equivalently, $2y = 2 imes 1 = 2$.

Notation and Multiplication Details

• Key rule: a number placed directly next to a variable implies multiplication. Examples: $2y = 2 imes y$, $3(x) = 3 imes x$, $4(y) = 4 imes y$.

• Why avoid the explicit multiplication symbol in early algebra: the symbol $\times$ can be confused with the variable name (especially with x).

• Common ways to write multiplication in algebra:

  • Juxtaposition: $2y$

  • Parentheses: $2(y)$ or $(2)y$

• The same expressions represent the same value regardless of notation used, as long as the meaning is clear.

Practice Connections and Takeaways

• Real-world relevance: algebra lets you model uncertain quantities (like tips or fluctuating prices) and compute outcomes by substituting values for the variables.

• Foundational principle: a variable stands for an unknown quantity; the constants (like 63) are fixed values in the scenario.

• Relationship between variables and expressions: changing a variable value changes the result of the expression, enabling exploration of how inputs influence outputs.

• Quick checks: always substitute the given value into the variable and simplify step by step to find the total.

Quick Practice Prompts

• If Mark’s tips on a day are $x = 5$, what is his total earnings? Show the substitution.

  • Answer outline: compute $63 + 5 = 68$.

• If on another day, x = 12, what is the earnings? Show the substitution.

  • Answer outline: compute $63 + 12 = 75$.

• If the price per gallon is $y = 0.50$ dollars, what is the total cost for two gallons?

  • Answer outline: $2y = 2 imes 0.50 = 1.00$ dollars.

• Why is $2y$ preferred over writing $2\times y$ in many algebra contexts?

  • Answer outline: to avoid confusion with the variable x and to keep notation clean and concise.

• Create your own two-variable model: a phone plan costs a fixed base of $a$ dollars plus each extra minute costs $b$ dollars. Write the expression for cost if you use $m$ extra minutes.

  • Answer outline: cost = $a + bm$.