Linear Expressions and First-Degree Equations

Course Preparation and Note-Taking Strategy

• Access lecture notes on Canvas prior to attending class sessions.

• Transcribe example problems into a spiral notebook in advance to ensure core content is ready prior to lecture instruction.

Evaluation of Algebraic Expressions

• Definition: Evaluating an algebraic expression requires substituting given numerical values for each variable in the expression and simplifying the resulting numeric expression using the standard order of operations.

• Order of Operations Order: Perform exponent operations first, followed by multiplication and division from left to right, and finally addition and subtraction from left to right.

• Example 1:

  • Expression: $6c - y$ where $c = 6$ and $y = -3$

  • Step 1 (Substitution): $6(6) - (-3)$

  • Step 2 (Multiplication): $6 \times 6 = 36$

  • Step 3 (Simplify double signs): $-(-3) = +3$

  • Step 4 (Addition): $36 + 3 = 39$

  • Result: $39$

• Example 2:

  • Expression: $m^2 + 4m - 5$ where $m = -7$

  • Step 1 (Substitution): $(-7)^2 + 4(-7) - 5$

  • Step 2 (Exponents): $(-7)^2 = (-7) \times (-7) = 49$

  • Step 3 (Multiplication): $4(-7) = -28$

  • Step 4 (Combine terms left to right): $49 - 28 - 5$

  • Step 5 (First subtraction): $49 - 28 = 21$

  • Step 6 (Final subtraction): $21 - 5 = 16$

  • Result: $16$

Simplification of Algebraic Expressions and Like Terms

• Definition of Like Terms: Like terms are terms that share the exact same variable(s) raised to the exact same exponent(s).

  • Terms with different variables (such as $x$ and $y$) cannot be combined.

  • Terms with different exponents on the same variable (such as $x$ and $x^2$) cannot be combined.

• Combining Method: Combine like terms by adding or subtracting their numerical coefficients while retaining the shared variable and exponent structure.

• Example 1:

  • Expression: $5(4w - 6) + 4(4w + 2)$

  • Step 1 (Distribute $5$): $5 \times 4w = 20w$, $5 \times (-6) = -30$

  • Step 2 (Distribute $4$): $4 \times 4w = 16w$, $4 \times 2 = 8$

  • Step 3 (Expanded expression): $20w - 30 + 16w + 8$

  • Step 4 (Group like terms): $(20w + 16w) + (-30 + 8)$

  • Step 5 (Combine coefficients): $(20 + 16)w = 36w$

  • Step 6 (Combine constants): $-30 + 8 = -22$ (When combining numbers of opposite signs, subtract the smaller absolute value from the larger absolute value $30 - 8 = 22$ and apply the sign of the larger absolute value, yielding $-22$).

  • Result: $36w - 22$

• Example 2:

  • Expression: $4y - 6(6z - y) - z$

  • Step 1 (Distribute $-6$): $-6 \times 6z = -36z$, $-6 \times (-y) = +6y$

  • Step 2 (Expanded expression): $4y - 36z + 6y - z$

  • Step 3 (Group $y$-terms): $4y + 6y = 10y$

  • Step 4 (Group $z$-terms): $-36z - 1z = -37z$

  • Result: $10y - 37z$

• Example 3:

  • Expression: $-8x^2 + 2 - 6x + 11x^2$

  • Step 1 (Group like terms): $-8x^2$ and $11x^2$ are like terms. The terms $-6x$ and $2$ have no matching like terms.

  • Step 2 (Combine $x^2$ terms): $-8 + 11 = 3 \rightarrow 3x^2$

  • Step 3 (Write in standard descending exponent form): $3x^2 - 6x + 2$

Foundations of Linear Equations in One Variable

• Definition: A linear equation in one variable is an equation that can be written in the standard form:   $ax + b = c$   where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

• First-Degree Equation: A linear equation is a first-degree equation, meaning the highest exponent on the variable is $1$

  • Example: $-5x - 9 = 14$ (Here, $a = -5$, $b = -9$, $c = 14$, and the degree of $x$ is $1$).

• Real Numbers: Includes all numbers that are not imaginary ($i$), encompassing positive integers, negative integers, fractions, decimals, and irrational numbers (e.g., repeating or non-terminating decimals).

• Solution: A real number value that, when substituted for the variable in an equation, simplifies both sides to form a true mathematical statement (e.g., $14 = 14$).

• Equivalent Equations: Equations that possess the exact same solution set.

Properties of Equality

• Addition Property of Equality:

  • Formal statement: If $a$, $b$, and $c$ are real numbers and $a = b$, then:     $a + c = b + c$

  • Adding or subtracting the same real number to both sides of an equation yields an equivalent equation.

  • Subtraction is equivalent to adding a negative number.

  • Application: Used to move constant or variable terms across the equal sign (e.g., adding $6$ to both sides of $4x - 6 = 25$).

• Multiplication Property of Equality:

  • Formal statement: If $a$, $b$, and $c$ are real numbers and $a = b$, then:     $a \times c = b \times c$

  • Multiplying or dividing both sides of an equation by the same non-zero real number yields an equivalent equation.

  • Division is equivalent to multiplying by a reciprocal fraction.

  • Application: Used to isolate variables by clearing coefficients (e.g., dividing both sides of $2x = 10$ by $2$ or multiplying by $\frac{1}{2}$).

Systematic Procedure for Solving Linear Equations

• Step 1: Clear Fractions or Decimals (Optional): Multiply every term on both sides of the equation by the Least Common Multiple (LCM) of all denominators (Least Common Denominator, LCD).

• Step 2: Clear Parentheses: Use the distributive property to eliminate all parentheses.

• Step 3: Combine Like Terms: Combine all like terms on the left side, then combine all like terms on the right side independently.

• Step 4: Isolate Variable Terms: Use addition or subtraction to move all terms containing the variable to one side of the equation.

• Step 5: Isolate Constant Terms: Use addition or subtraction to move all constant terms to the opposite side of the equation.

• Step 6: Solve for the Variable: Use multiplication or division to set the coefficient of the variable to $1$

• Step 7: Check Solution (Optional): Substitute the calculated solution back into the original, unmodified equation to confirm it yields a true identity statement.

Step-by-Step Problem Solutions

• Problem 1:

  • Equation: $4w + 8 = 15w - 3w + 64$

  • Step 1 (Combine like terms on right): $15w - 3w = 12w$

  • Equation simplified: $4w + 8 = 12w + 64$

  • Step 2 (Move variable terms): Subtract $4w$ from both sides:     $8 = 8w + 64$

  • Step 3 (Move constant terms): Subtract $64$ from both sides:     $8 - 64 = -56$     $-56 = 8w$

  • Step 4 (Isolate variable): Divide both sides by $8$     $w = \frac{-56}{8} = -7$

  • Solution: $w = -7$

• Problem 2:

  • Equation: $4v + 20 = -4(v + 7)$

  • Step 1 (Distribute $-4$): $-4 \times v = -4v$ and $-4 \times 7 = -28$

  • Equation simplified: $4v + 20 = -4v - 28$

  • Step 2 (Move variable terms): Add $4v$ to both sides:     $8v + 20 = -28$

  • Step 3 (Move constant terms): Subtract $20$ from both sides:     $8v = -28 - 20 = -48$

  • Step 4 (Isolate variable): Divide both sides by $8$     $v = \frac{-48}{8} = -6$

  • Verification:

  • Left Side: $4(-6) + 20 = -24 + 20 = -4$

  • Right Side: $-4(-6 + 7) = -4(1) = -4$

  • Statement $-4 = -4$ is true.

  • Solution: $v = -6$

• Problem 3:

  • Equation: $2(x - 4) - 2 = 3(7x - 2) - 9x$

  • Step 1 (Distribute):

  • Left side: $2(x - 4) - 2 = 2x - 8 - 2$

  • Right side: $3(7x - 2) - 9x = 21x - 6 - 9x$

  • Step 2 (Combine like terms):

  • Left side: $-8 - 2 = -10 \rightarrow 2x - 10$

  • Right side: $21x - 9x = 12x \rightarrow 12x - 6$

  • Equation simplified: $2x - 10 = 12x - 6$

  • Step 3 (Move variable terms): Subtract $2x$ from both sides:     $-10 = 10x - 6$

  • Step 4 (Move constant terms): Add $6$ to both sides:     $-10 + 6 = -4$     $-4 = 10x$

  • Step 5 (Isolate variable): Divide both sides by $10$     $x = \frac{-4}{10}$

  • Step 6 (Reduce fraction): Divide numerator and denominator by $2$     $x = -\frac{2}{5}$

  • Solution: $x = -\frac{2}{5}$

• Problem 4 (Clearing Fractional Coefficients):

  • Equation: $\frac{-5}{2} - \frac{7}{3}x = \frac{-9}{5}$

  • Step 1 (Determine LCD):

  • Denominators: $2$, $3$, $5$

  • Multiples of $5$: $5, 10, 15, 20, 25, 30$

  • Least Common Multiple ($\text{LCD}$) = $30$ ($2 \times 3 \times 5 = 30$)

  • Step 2 (Multiply every term by $30$):

  • First term: $30 \times \frac{-5}{2} = \frac{-150}{2} = -75$

  • Second term: $30 \times \frac{-7}{3}x = \frac{-210}{3}x = -70x$

  • Third term: $30 \times \frac{-9}{5} = \frac{-270}{5} = -54$       (Diagonal simplification technique: $30 \times \frac{-9}{5} = 6 \times (-9) = -54$)

  • Cleared equation: $-75 - 70x = -54$

  • Step 3 (Move constant terms): Add $75$ to both sides:     $-70x = -54 + 75$     $-70x = 21$

  • Step 4 (Isolate variable): Divide both sides by $-70$     $x = \frac{21}{-70}$

  • Step 5 (Reduce fraction): Divide numerator and denominator by $7$     $x = -\frac{3}{10}$

  • Solution: $x = -\frac{3}{10}$

• Problem 5 (Clearing Fractional Coefficients):

  • Equation: $\frac{4}{3}v - 5 = \frac{4}{5}v + \frac{1}{3}$

  • Step 1 (Determine LCD):

  • Unique denominators: $3$, $5$

  • Least Common Multiple ($\text{LCD}$) = $15$ ($3 \times 5 = 15$)

  • Step 2 (Multiply every term by $15$):

  • First term: $15 \times \frac{4}{3}v = \frac{60}{3}v = 20v$

  • Second term: $15 \times (-5) = -75$

  • Third term: $15 \times \frac{4}{5}v = \frac{60}{5}v = 12v$

  • Fourth term: $15 \times \frac{1}{3} = \frac{15}{3} = 5$

  • Cleared equation: $20v - 75 = 12v + 5$

  • Step 3 (Move variable terms): Subtract $12v$ from both sides:     $8v - 75 = 5$

  • Step 4 (Move constant terms): Add $75$ to both sides:     $8v = 80$

  • Step 5 (Isolate variable): Divide both sides by $8$     $v = 10$

  • Solution: $v = 10$

Questions & Discussion

• Question on Ordering Terms:

  • Question: On tests and quizzes, is strict adherence to placing terms in descending exponent order (with variable terms first and constant terms at the end) required?

  • Answer: At the introductory stage of solving linear equations, term order flexibility is allowed and non-descending order will not result in point deductions. However, as advanced algebraic topics progress, organizing expressions in standard polynomial order (highest degree to lowest degree) becomes standard expectation.