Solving One-Step Linear Equations

Fundamentals of One-Step Linear Equations

• Definition of an Equation: An algebraic statement that contains an equal sign ($=$). An equation is always defined by the presence of an equal sign ($=$).

• Definition of an Unknown (Variable): A letter (such as $x$) that represents an unknown numeric value.

• Goal of Solving an Equation: To determine the exact value of the unknown variable ($x$).

• Structure and Visualizing Equations:

  • An equation can be visualized as being split down the middle by a line passing directly through the equal sign ($=$).
  • This split creates two distinct sides: the Left Hand Side (LHS) and the Right Hand Side (RHS).

• The Fundamental Rule of Equations: Whatever operation is performed on one side of the equation must also be performed on the other side to maintain balance.

• Methodology for Isolating Variables:

  • To solve for $x$, the term containing $x$ must be isolated on one side so that only $1x$ (or $x$) remains.
  • Equal signs should be aligned vertically from step to step.
  • To eliminate extra terms or coefficients, apply the inverse (opposite) operation:
    • The inverse of addition ($+$) is subtraction ($-$).
    • The inverse of subtraction ($-$) is addition ($+$).
    • The inverse of multiplication ($\times$) is division ($\text{/}$ or $\frac{}{}$).
    • The inverse of division ($\text{/}$ or $\frac{}{}$) is multiplication ($\times$).

Solving Basic One-Step Equations

• Addition Equations:

  • Example Equation: $x + 3 = 7$
  • LHS Terms: $x$ and $+3$.
  • Inverse Operation: The inverse of adding $3$ is subtracting $3$.
  • Step-by-Step Solution:
    • Subtract $3$ from both the LHS and RHS.
    • $\text{LHS}: 3 - 3 = 0$ (cancels out the $+3$, leaving only $x$).
    • $\text{RHS}: 7 - 3 = 4$.
    • $\text{Result}: x = 4$

• Subtraction Equations:

  • Example Equation: $x - 8 = 1$
  • LHS Terms: $x$ and $-8$.
  • Inverse Operation: The inverse of subtracting $8$ is adding $8$.
  • Step-by-Step Solution:
    • Add $8$ to both the LHS and RHS.
    • $\text{LHS}: -8 + 8 = 0$ (cancels out the $-8$, leaving only $x$).
    • $\text{RHS}: 1 + 8 = 9$.
    • $\text{Result}: x = 9$

• Multiplication Equations:

  • Example Equation: $4x = 12$
  • Meaning of Terms: $4x$ represents $4 \times x$ ($4$ multiplied by $x$).
  • Inverse Operation: The inverse of multiplying by $4$ is dividing by 4$.\n * **Step-by-Step Solution:**\n * Divide both the LHS and RHS by 4$.
    • $\text{LHS}: 4x \text{ divided by } 4 = 1x$ (or $x$).
    • $\text{RHS}: 12 \text{ divided by } 4 = 3$.
    • $\text{Result}: x = 3$

• Division Equations:

  • Example Equation: $\frac{x}{2} = 10$
  • Meaning of Terms: $\frac{x}{2}$ represents $x$ divided by 2$.\n * **Inverse Operation:** The inverse of dividing by 2$is multiplying by$2$.
  • Step-by-Step Solution:
    • Multiply both the LHS and RHS by 2$.\n * \text{LHS}: (x \text{ divided by } 2) \times 2 = x$(division and multiplication by$2 cancel out).\n * \text{RHS}: 10 \times 2 = 20.\n * \text{Result}: x = 20\n\n# Advanced Examples and Special Cases\n\n* **Case 1: Subtraction Resulting in a Negative Solution**\n * **Example Equation:** x + 7 = 2\n * **Step-by-Step Solution:**\n * Subtract 7$from both sides to eliminate$+7 from the LHS.\n * \text{LHS}: +7 - 7 = 0$, leaving$x.\n * \text{RHS}: 2 - 7 = -5\n * \text{Result}: x = -5\n\n* **Case 2: Division of a Negative Integer**\n * **Example Equation:** 6x = -12\n * **Step-by-Step Solution:**\n * Divide both sides by 6$to eliminate multiplication by$6.\n * \text{LHS}: 6x \text{ divided by } 6 = 1x\n * \text{RHS}: -12 \text{ divided by } 6 = -2\n * \text{Result}: x = -2\n\n* **Case 3: Variable Located on the Right Hand Side**\n * **Example Equation:** 10 = x + 4\n * **Step-by-Step Solution:**\n * Isolate x$on the RHS by applying the inverse of$+4$, which is subtracting$4 from both sides.\n * \text{LHS}: 10 - 4 = 6\n * \text{RHS}: +4 - 4 = 0$, leaving$x\n * \text{Intermediate Result}: 6 = x\n * **Standard Practice:** Conventionally, the variable is written on the LHS. Reverse the statement 6 = x to express the final answer in standard form.\n * \text{Final Result}: x = 6\n\n* **Case 4: Solutions Resulting in Simplified Fractions**\n * **Example Equation:** 8x = 2\n * **Step-by-Step Solution:**\n * Divide both sides by 8$to isolate$x$.
    • $\text{LHS}: 8x \text{ divided by } 8 = 1x$
    • $\text{RHS}: 2 \text{ divided by } 8$
  • Avoiding Common Errors:
    • Do not confuse $2 \text{ divided by } 8$ with $8 \text{ divided by } 2$. While $8 \text{ divided by } 2 = 4$, $2 \text{ divided by } 8$ is a fraction less than $1$.
  • Writing and Simplifying Fractional Results:
    • Express $2 \text{ divided by } 8$ as the fraction $\frac{2}{8}$.
    • Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, $2$:
      • $\text{Numerator}: 2 \text{ divided by } 2 = 1$
      • $\text{Denominator}: 8 \text{ divided by } 2 = 4$
    • $\text{Final Result}: x = \frac{1}{4}$