08/21/25

Setup and class context

• The class emphasizes homework with due dates and a preference to work through topics in class, then review at home.

• Instructor uses Math Lab for assignments with due dates shown on the assignment page.

• Videos exist for each assignment (publisher videos) but are not required; instructor records own class videos for review.

• You may be allowed to rewatch class videos at home to review what was said; the morning class video is shared for reference.

• Emphasis on working through the current section (1.7 in this transcript) with problems that may have different numbers but the same underlying structure.

• The instructor encourages calculator use for both simple and negative-number arithmetic to reduce frustration and errors.

• If you miss concepts during class, you can rewatch videos or email the instructor for clarification; there is flexibility in how you submit answers in Math Lab, but some cases are picky.

Key concepts: terms, coefficients, and variable parts

• A term contains no addition or subtraction; it is made up of multiplication/division components.

• The number in front of a term is the coefficient.

• What follows the coefficient is the variable part (the variables and their exponents).

• Variable parts can include exponents on each variable, e.g., $x^2 y^3$ or $x^3 y^2$.

• Like terms are terms whose variable parts are identical (same variables with the same exponents).

• Terms with the same variable but different exponents are not like terms.

• Constants are terms with no variable part (just numbers).

• Examples:

  • Are $7r$ and $-11r$ like terms? Yes (variable part is $r$).

  • Are $4a^3$ and $6a^2$ like terms? No (exponents on the same variable $a$ differ).

  • Are $h$ and $k$ like terms? No (different letters).

• Important terms:

  • Coefficient: the number in front of the term (e.g., in $-2r$, the coefficient is $-2$).

  • Variable part: the letters and their exponents (e.g., in $-2r$, the variable part is $r$).

  • Exponents: the numbers to the right of variables (e.g., in $x^2 y^3$, exponents are 2 on x and 3 on y).

Identifying and working with like terms

• To determine if two terms are like terms, compare their variable parts only (not coefficients).

• If the variable parts are identical, the terms are like terms; you can add/subtract their coefficients.

• If the variable parts are not identical, you cannot add or subtract the terms.

• Examples:

  • $7r + (-11)r$ -> like terms; combine coefficients: $7 + (-11) = -4$, giving $-4r$.

  • $4a^3 + 6a^2$ -> not like terms; cannot combine.

  • $h + k$ -> not like terms; cannot combine.

• When combining like terms with a common variable, keep the common variable part and add/subtract the coefficients.

• Special note on coefficients:

  • If a coefficient is not written, it is understood to be 1 (e.g., the coefficient of $x$ in $x$ is $1$, and the coefficient of $y^2$ in $y^2$ is $1$).

Examples: combining like terms (worked walkthroughs)

• Example 1: $-6s + 1s$

  • Variable part: $s$ for both terms -> like terms

  • Coefficients: $-6 + 1 = -5$

  • Result: $-5s$

• Example 2: $14x + x$

  • Coefficients: $14 + 1 = 15$ (since the second term has an implicit coefficient of $1$)

  • Result: $15x$

• Example 3: combine a constant and a multiple-term

  • Expression: $5t + 20 - 15$

  • Only term with $t$ is $5t$ (no other like terms to combine with $t$)

  • Combine constants: $20 - 15 = 5$

  • Result: $5t + 5$

• Example 4: combining multiple terms with a common variable

  • Expression: $7x + 5x + 4 + 3$

  • Like terms with $x$: $7x + 5x = 12x$

  • Constants: $4 + 3 = 7$

  • Final: $12x + 7$

• Important practice tips:

  • Keep a clear separation between terms; use plus/minus signs to separate terms.

  • If a term has a coefficient of 1, you can omit the 1 in writing (but do not omit in calculations where clarity is needed).

  • Always check if the remaining constants can be combined after like-term combination.

The order of operations (PEMDAS) and parentheses

• PEMDAS stands for:

  • P: Parentheses

  • E: Exponents

  • MD: Multiplication and Division (from left to right)

  • AS: Addition and Subtraction (from left to right)

• When dealing with parentheses:

  • Simplify inside parentheses first.

  • Inside a parenthesis that contains addition/subtraction, apply the order of operations within it.

• Distribution (FOIL-like concept for multiplication over a parenthesized expression):

  • Example: $3 imes (s - 3t)$

  • Distribute 3 to each term inside: $3 imes s = 3s$, $3 imes (-3t) = -9t$

  • Result: $3s - 9t$

  • Negative distribution: $-2 imes (t - 3)$

  • Distribute -2: $-2t + 6$

• Additional notes:

  • Signs matter: same signs multiply to positive; different signs multiply to negative.

  • There is a distinction between commutativity with addition and subtraction:

  • You can rearrange terms when adding (e.g., $a + b = b + a$) and keep the signs with the correct terms.

  • Subtraction is not commutative in the same way; you cannot arbitrarily swap signs between terms that involve subtraction.

• Example walkthroughs in class included several problems where instructors showed step-by-step application of distribution and combining like terms to simplify expressions.

Evaluating expressions via substitution

• When evaluating expressions, replace each variable with a given number and simplify using the order of operations.

• Substitution technique:

  • Use extra parentheses around the substitution to avoid ambiguity, especially when the substituted value is negative or when there are multiple operations.

  • Example structure: if evaluating $7x + 2y + 6$ with $x = 4$ and $y = 3$:

  • Substitution: replace x with 4 and y with 3 → write as $7(4) + 2(3) + 6$

  • Apply MDAS: multiplication first, then addition: $28 + 6 + 6 = 40$

• Two-variable example: $7x + 2y + 6$ with $x=4, y=3$ → $40$ (as shown above).

• Example: $6x - 4y + 4a$ with $x=5, y=-7, a=6$:

  • Compute step-by-step with order:

  • $6(5) - 4(-7) + 4(6) = 30 + 28 + 24 = 82$

• Example: $-4y^2 + 9a^2$ with $y=-7, a=4$:

  • Exponents first: $(-7)^2 = 49, 4^2 = 16$

  • Then multiply: $-4 imes 49 = -196$, $9 imes 16 = 144$

  • Sum: $-196 + 144 = -52$

• Example: $rac{7y^2 + 2}{x}$ with $x=6, y=-4$:

  • Compute numerator first: $7(-4)^2 + 2 = 7(16) + 2 = 112 + 2 = 114$

  • Then divide by x: $114 / 6 = 19$

• Example: Mixed substitution with a more complex expression (from transcript): evaluate

  • $6x - 4y + 4a$ with values $x=5, y=-7, a=6$ gives $82$ (as shown).

• Common mistakes highlighted:

  • Do not perform multiplication before addressing exponents inside a term; exponents take precedence over multiplication when evaluating, e.g., for a term like $2 imes 4^2$ you must compute $4^2 = 16$ first, then multiply by 2 to get $32$, not the other way around.

  • When substituting, if you skip expanding parenthesis or mis-handle signs, you can get incorrect results; the instructor emphasizes checking the order of operations carefully.

Practical tips and common pitfalls

• Always perform exponents before multiplication/division, and perform multiplication/division before addition/subtraction.

• When combining like terms, ensure the variable parts match exactly (same variables with same exponents).

• If you are unsure about a calculator step or negative number handling, rely on the calculator to verify basic arithmetic to reduce frustration.

• Use parentheses deliberately when substituting values to keep the intended grouping clear (especially when variables are part of larger expressions).

• In graded settings (Math Lab), answer formats may be flexible to some extent, but be prepared to adjust if the platform requires a specific form.

• When reviewing homework that you didn’t finish in class, you can rewatch the class video and rework problems to reinforce understanding before the due date.

• Save progress frequently in the homework system so you can return and retry problems before the deadline.

Quick practice scenarios (summary problems to reinforce concepts)

• Determine if the following are like terms:

  • $7r$ and $-11r$ → like terms (variable parts identical: $r$)

  • $4a^3$ and $6a^2$ → not like terms (exponents differ)

  • $h$ and $k$ → not like terms (different letters)

• Combine like terms:

  • $-6s + 1s = -5s$

  • $14x + x = 15x$

  • $5t + 20 - 15 = 5t + 5$

• Distribute over parentheses:

  • $3(s - 3t) = 3s - 9t$

  • $-2(t - 3) = -2t + 6$

• Substitution practice (single variable): if $f(x) = 13 + x$ and $x = 8$, then $f(8) = 21$; practice with grouping for clarity.

• Substitution practice (two variables): evaluate $7x + 2y + 6$ with $x = 4, y = 3$ → $40$.

• Multi-term substitution with negatives: evaluate $6x - 4y + 4a$ with $x=5, y=-7, a=6$ → $82$; evaluate a second example with negatives and squares to reinforce exponent rules.

• Evaluating a fractional expression with substitution: evaluate

  • $rac{7y^2 + 2}{x}$ with $y = -4, x = 6$ → $rac{114}{6} = 19$

• Takeaway: always follow the order of operations strictly, use substitution carefully, and check your steps with a calculator when needed.

Encouragement and wrap-up

• The material covered (like terms, combining like terms, distribution, order of operations, and evaluating expressions) is foundational and will be used throughout the semester.

• Keep practicing to build fluency, especially with negative numbers and exponents.

• If you have questions, ask during class or email the instructor for clarification; you can also review the morning-class video for additional explanations.

• Remember: one small mistake can change the result, so slow, careful steps and consistent use of parentheses help prevent errors.