Grade 7 Mathematics: Algebraic Expressions (Bentuk Aljabar)

Fundamentals of Algebraic Expressions

• Algebraic expressions in Grade 7 Mathematics (under the Merdeka Curriculum / Kurikulum Merdeka) are constructed from four foundational elements:

  • Terms ($\text{suku}$
  • Coefficients ($\text{koefisien}$
  • Variables ($\text{variabel}$
  • Constants ($\text{konstanta}$

• A term ($\text{suku}$) is defined as an algebraic entity separated by fundamental arithmetic operational symbols, specifically addition, subtraction, multiplication, and division:

  • A single-term expression (monomial) example is $2x$, which consists of exactly $1$ term.
  • A two-term expression (binomial) example is $2x + 4$, which consists of $2$ terms: $2x$ and $4$.
  • A three-term expression (trinomial) example is $2a + 2b + 5$, which consists of $3$ terms: $2a$, $2b$, and $5$.

• A coefficient ($\text{koefisien}$) is defined as the numerical value that precedes and multiplies a variable.

• A variable ($\text{variabel}$) is defined as the letter representing an unknown quantity within an algebraic expression.

• A constant ($\text{konstanta}$) is defined as a standalone numerical value that is not followed by or multiplied by a letter/variable.

• Comprehensive structural breakdown of the expression $2x + 4$:

  • The component $2x$ functions as a term.
  • The component $4$ functions as a term.
  • Isolating the number $2$ identifies it specifically as the coefficient of the variable $x$.
  • Isolating the letter $x$ identifies it specifically as the variable.
  • Isolating the standalone number $4$ identifies it specifically as the constant.

Addition of Algebraic Expressions

• The governing rule of algebraic addition dictates that expressions can only be solved or combined if they contain identical variables (like terms).

• If terms possess differing variables, they cannot be simplified further and must remain written separately.

• In the addition operation, only the numerical coefficients are summed together, while the variable remains completely unchanged.

• Direct addition examples:

  • In the expression $2a + 3a$:
  • Add the numerical coefficients: $2 + 3 = 5$.
  • Retain the variable $a$.
  • Final simplified sum: $5a$.
  • In the expression $3b + 4b$:
  • Add the numerical coefficients: $3 + 4 = 7$.
  • Retain the variable $b$.
  • Final simplified sum: $7b$.

• Multi-variable addition procedure for $3a^2 + 5b + 3b + 4a^2$:

  • Step 1: Identify all variable types present ($a^2$ and $b$).
  • Step 2: Reorder and group like terms together: $3a^2 + 4a^2 + 5b + 3b$.
  • Step 3: Combine terms sharing the variable $a^2$ by adding their coefficients: $3 + 4 = 7$, resulting in $7a^2$.
  • Step 4: Recognize that $7a^2$ cannot be added to the $b$ terms because the variables are distinct.
  • Step 5: Combine terms sharing the variable $b$ by adding their coefficients: $5 + 3 = 8$, resulting in $8b$.
  • Step 6: Combine the simplified distinct groups into the final irreducible expression: $7a^2 + 8b$.

Subtraction of Algebraic Expressions

• Identical to addition, algebraic subtraction can only be performed across terms sharing the exact same variables.

• Direct subtraction examples:

  • In the expression $5a - 3a$:
  • Subtract the coefficients: $5 - 3 = 2$.
  • Retain the variable $a$.
  • Final result: $2a$.
  • In the expression $7c^2 - 5c^2$:
  • Subtract the coefficients: $7 - 5 = 2$.
  • Retain the variable component $c^2$.
  • Final result: $2c^2$.
  • In the expression $4b - 3b$:
  • Subtract the coefficients: $4 - 3 = 1$.
  • By convention, a coefficient of $1$ is omitted in writing.
  • Final result: $b$.

• Properties of implicit coefficients and operational edge cases:

  • When an algebraic variable appears without an explicit preceding number, its coefficient is implicitly $1$.
  • Subtraction involving a lone variable, such as $7a - a$:
  • The variable $a$ does not disappear.
  • The term $-a$ has an implicit coefficient of $-1$.
  • Subtract the coefficients: $7 - 1 = 6$.
  • Retain the variable component to yield $6a$.
  • Complete cancellation scenario, such as $a - a$:
  • Both terms possess an implicit coefficient of $1$.
  • Subtract the coefficients: $1 - 1 = 0$.
  • The product of $0$ and any variable is $0$ ($0a = 0$), meaning the variable is dropped entirely.
  • Final result: $0$.

Multiplication of Algebraic Expressions

• The governing rule of algebraic multiplication requires multiplying both the numerical coefficients and the variables across all terms involved.

• Fundamental multiplication rules:

  • Multiplying distinct variables concatenates them: $a \times b = ab$.
  • Multiplying a scalar number and a variable concatenates them with the number written first: $2 \times a = 2a$.

• Sequential and monomial multiplication examples:

  • Multiplying a monomial term by a scalar: $2a \times 4$
  • Multiply the numerical coefficients: $2 \times 4 = 8$.
  • Retain the single variable $a$.
  • Final product: $8a$.
  • Multiplying terms with distinct variables: $3a \times 2b$
  • Multiply the coefficients: $3 \times 2 = 6$.
  • Multiply the variables: $a \times b = ab$.
  • Final product: $6ab$.
  • Multiplying terms with identical variables: $3a \times a$
  • The variable $a$ has an implicit coefficient of $1$, so the numerical calculation is $3 \times 1 = 3$.
  • Multiplying identical variables increases the exponent rather than yielding a coefficient: $a \times a = a^2$ (not $2a$).
  • Final product: $3a^2$.
  • Multiplying multiple identical variables sequentially: $3a \times 4a \times 5a$
  • Multiply all numerical coefficients in sequence: $3 \times 4 = 12$, followed by $12 \times 5 = 60$.
  • Multiply the variable factors: $a \times a \times a = a^3$.
  • The exponent value corresponds directly to the total number of identical variable factors being multiplied ($3$ factors of $a$ yield an exponent of $3$).
  • Final product: $60a^3$.

• Distributive property of multiplication over addition and subtraction:

  • Expressions enclosed within parentheses denote multiplication; every term located outside the parentheses must be multiplied by every term inside.
  • Monomial distributed across a binomial: $3(a + b)$
  • Multiply $3$ by the first term $a$: $3 \times a = 3a$.
  • Multiply $3$ by the positive second term $b$: $3 \times (+b) = +3b$.
  • Final product: $3a + 3b$.
  • Binomial multiplied by a binomial: $(a + 3)(a - 4)$
  • Every component in the first set of parentheses must be distributed to every component in the second set of parentheses.
  • Step 1: Multiply the first term of the first binomial by the first term of the second binomial: $a \times a = a^2$.
  • Step 2: Multiply the first term of the first binomial by the second term of the second binomial (retaining the negative sign): $a \times (-4) = -4a$.
  • Step 3: Multiply the second term of the first binomial by the first term of the second binomial: $+3 \times a = +3a$.
  • Step 4: Multiply the second term of the first binomial by the second term of the second binomial: a positive multiplied by a negative yields a negative result ($3 \times (-4) = -12$).
  • Step 5: Write out the expanded intermediate polynomial: $a^2 - 4a + 3a - 12$.
  • Step 6: Combine the like terms in the center: $-4a + 3a = -a$.
  • Final simplified expression: $a^2 - a - 12$.

Division of Algebraic Expressions

• The governing rule of algebraic division requires dividing numerical coefficients by numerical coefficients and dividing variables by matching variables.

• Fractional bars ($\frac{\text{numerator}}{\text{denominator}}$) denote standard division operations.

• Direct division and exponent subtraction rules:

  • When dividing identical variables, the powers are subtracted according to the exponent quotient rule: if the base variable is the same, subtract the exponent of the denominator from the exponent of the numerator.
  • Any variable written without an explicit exponent has an implicit exponent of $1$ ($a = a^1$).

• Single-variable division examples:

  • Division with identical terms: $6a \div 2a$
  • Divide the coefficients: $6 \div 2 = 3$.
  • Divide matching variables: $a \div a = 1$, meaning the variable cancels out completely ($\frac{1}{1} = 1$).
  • Final quotient: $3$.
  • Division with unequal powers: $7a^2 \div 7a$
  • Divide the coefficients: $7 \div 7 = 1$ (an initial coefficient of $1$ is omitted in standard notation).
  • Apply exponent subtraction to the variable base $a$: $a^2 \div a^1 = a^{2 - 1} = a^1$.
  • Because a power of $1$ is not explicitly written, $a^1$ is written simply as $a$.
  • Final quotient: $a$.

• Multi-variable algebraic fraction simplification examples:

  • Simplifying $\frac{6ab^2}{3a}$:
  • Divide the numerical coefficients: $6 \div 3 = 2$.
  • Cancel the variable $a$ present in both the numerator and denominator ($a \div a = 1$).
  • Retain the remaining variable $b^2$ in the numerator.
  • Final simplified expression: $2b^2$.
  • Simplifying $\frac{12a^5b^4c^3}{6a^3b^2c}$:
  • Divide the numerical coefficients: $12 \div 6 = 2$.
  • Divide matching variable base $a$ by subtracting exponents: $a^5 \div a^3 = a^{5 - 3} = a^2$.
  • Divide matching variable base $b$ by subtracting exponents: $b^4 \div b^2 = b^{4 - 2} = b^2$.
  • Divide matching variable base $c$ by subtracting exponents ($c = c^1$): $c^3 \div c^1 = c^{3 - 1} = c^2$.
  • Consolidate all quotient terms into the final simplified expression: $2a^2b^2c^2$.