Linear Equations in One Variab

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### Introduction to Linear Equations A **linear equation** in one variable is a mathematical statement that two expressions are equal, where one variable (e.g., $x$) is raised to the power of 1. It's used to find an unknown value. The core concept is a **balance scale**: - The **equals sign** ($=$) is the center. - The **Left-Hand Side (LHS)** must equal the **Right-Hand Side (RHS)**. - Any operation on one side must be performed on the other to maintain balance. #### Anatomy of an Equation For $3x + 5 = 20$: - **Variable ($x$):** The unknown value. - **Coefficient ($3$):** Number multiplying the variable. - **Constants ($5, 20$):** Fixed numerical values. - **Operator ($+$):** Mathematical action (addition, subtraction, multiplication, division). ### The Golden Rule & Goal #### The Golden Rule > **Whatever you do to one side of the equation, you MUST do to the other side.** This ensures the equation remains balanced. #### The Goal: Isolation Your objective is to **isolate the variable** ($x$) to find its value: $$ x = \text{some number} $$ ### Toolkit: Inverse Operations To isolate the variable, use **inverse operations** to "undo" what's been done to $x$: - **Addition ($+$)** $\leftrightarrow$ **Subtraction ($-$)** - **Multiplication ($\times$)** $\leftrightarrow$ **Division ($\div$)** ### Solving Equations Step-by-Step #### 1. One-Step Equations Require a single inverse operation. **Example:** Solve for $x$: $x - 7 = 12$ - **Action:** Add 7 to both sides (inverse of subtracting 7). $$ \begin{aligned} x - 7 \mathbf{+ 7} &= 12 \mathbf{+ 7} \\ x &= 19 \end{aligned} $$ **Check:** $19 - 7 = 12$ (True) #### 2. Two-Step Equations 1. Undo addition/subtraction (move constant). 2. Undo multiplication/division (move coefficient). **Example:** Solve for $x$: $4x + 6 = 26$ - **Step 1 (Undo addition):** Subtract 6 from both sides. $$ \begin{aligned} 4x + 6 \mathbf{- 6} &= 26 \mathbf{- 6} \\ 4x &= 20 \end{aligned} $$ - **Step 2 (Undo multiplication):** Divide by 4 on both sides. $$ \begin{aligned} \frac{4x}{\mathbf{4}} &= \frac{20}{\mathbf{4}} \\ x &= 5 \end{aligned} $$ **Check:** $4(5) + 6 = 20 + 6 = 26$ (True) #### 3. Variables on Both Sides Move all variable terms to one side, and all constant terms to the other. **Example:** Solve for $x$: $5x - 3 = 3x + 9$ - **Step 1 (Move variables):** Subtract $3x$ from both sides. $$ \begin{aligned} 5x - 3 \mathbf{- 3x} &= 3x + 9 \mathbf{- 3x} \\ 2x - 3 &= 9 \end{aligned} $$ - **Step 2 (Move constants):** Add 3 to both sides. $$ \begin{aligned} 2x - 3 \mathbf{+ 3} &= 9 \mathbf{+ 3} \\ 2x &= 12 \end{aligned} $$ - **Step 3 (Isolate $x$):** Divide by 2 on both sides. $$ \begin{aligned} \frac{2x}{\mathbf{2}} &= \frac{12}{\mathbf{2}} \\ x &= 6 \end{aligned} $$ **Check:** $5(6) - 3 = 30 - 3 = 27$ and $3(6) + 9 = 18 + 9 = 27$ (True) ### Procedure Summary 1. **Simplify:** Distribute to remove parentheses; combine like terms on the same side. 2. **Collect Variables:** Use addition/subtraction to move all variable terms to one side. 3. **Collect Constants:** Use addition/subtraction to move all constant terms to the other side. 4. **Isolate:** Use multiplication/division to remove the coefficient from the variable. 5. **Verify:** Substitute your solution back into the *original* equation to confirm. ### Real-World Application **Scenario:** A streaming service charges a $\$15$ setup fee and $\$10$ per month. You have $\$75$. How many months ($m$) can you subscribe? **Equation:** $10m + 15 = 75$ **Solve:** 1. Subtract 15 (setup fee) from both sides: $$ \begin{aligned} 10m + 15 \mathbf{- 15} &= 75 \mathbf{- 15} \\ 10m &= 60 \end{aligned} $$ 2. Divide by 10 (monthly cost) on both sides: $$ \begin{aligned} \frac{10m}{\mathbf{10}} &= \frac{60}{\mathbf{10}} \\ m &= 6 \end{aligned} $$ **Answer:** You can subscribe for 6 months.