Solving Linear Equations in One Variable

TL;DR

Solving linear equations means finding the specific value for the unknown variable that makes the equation true. You do this by isolating the variable on one side of the equals sign using inverse operations. Think of the equation like a balanced scale; whatever you do to one side, you must do to the other to keep it balanced.

1. The Mental Model

Imagine an equation as a perfectly balanced seesaw. Your goal is to figure out how much weight (the variable) is on one side. To do that, you need to get that weight all by itself, but every time you add or remove something from one side, you must do the exact same thing to the other side to keep it balanced.

2. The Core Material

When you're solving linear equations with one variable, you're looking for the single value that makes the equation true. The key is to isolate the variable, meaning you want to get it by itself on one side of the equals sign. You achieve this by performing inverse operations.

Understanding Inverse Operations

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Inverse operations "undo" each other.
* Addition and Subtraction are inverse operations.
* Multiplication and Division are inverse operations.

The Balancing Act

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The most crucial rule is: Whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced and ensures the solution remains valid.

Let's break down the general steps:

graph TD
    Start["Start with the equation"] --> Simplify["Simplify both sides (distribute, combine like terms)"]
    Simplify --> VariableOnOneSide["Move all terms with the variable to one side"]
    VariableOnOneSide --> ConstantOnOtherSide["Move all constant terms to the other side"]
    ConstantOnOtherSide --> IsolateVariable["Isolate the variable (multiply/divide to get 'x = ...')"]
    IsolateVariable --> CheckSolution["(Optional) Check your solution by plugging it back in"]
    CheckSolution --> End["End"]

Step-by-Step Approach

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1. Simplify Both Sides: Look at each side of the equation separately. Distribute any numbers outside parentheses and combine any like terms (e.g., $3x + 2x$ becomes $5x$, or $5 - 2$ becomes $3$).
2. Move Variable Terms: Use addition or subtraction to get all terms containing the variable on one side of the equation (usually the left) and all constant terms on the other side (usually the right).
3. Isolate the Variable: Once you have a single variable term (like $5x$) and a single constant term (like $15$), use multiplication or division to get the variable completely by itself. If you have $5x = 15$, you'd divide both sides by $5$.
4. Check Your Answer (Optional but Recommended!): Plug the value you found for the variable back into the original equation. If both sides are equal, your solution is correct.

3. Worked Example

Let's solve the equation: $3(x - 2) + 5x = 10 + 2x$

1. Simplify Both Sides:

   • Left side: Distribute the 3: $3x - 6 + 5x$. Combine like terms: $8x - 6$.
   • Right side: No simplifying needed: $10 + 2x$.
   • Equation becomes: $8x - 6 = 10 + 2x$

2. Move Variable Terms: We want all $x$ terms on one side. Let's subtract $2x$ from both sides.

   • $(8x - 6) - 2x = (10 + 2x) - 2x$
   • $6x - 6 = 10$

3. Move Constant Terms: Now, let's get the numbers without $x$ on the other side. Add $6$ to both sides.

   • $(6x - 6) + 6 = 10 + 6$
   • $6x = 16$

4. Isolate the Variable: The $x$ is being multiplied by $6$. To undo this, divide both sides by $6$.

   • $\frac{6x}{6} = \frac{16}{6}$
   • $x = \frac{16}{6}$
   • Simplify the fraction: $x = \frac{8}{3}$

5. Check Your Answer: Substitute $x = \frac{8}{3}$ back into the original equation:

   • $3(\frac{8}{3} - 2) + 5(\frac{8}{3}) = 10 + 2(\frac{8}{3})$
   • $3(\frac{8}{3} - \frac{6}{3}) + \frac{40}{3} = 10 + \frac{16}{3}$
   • $3(\frac{2}{3}) + \frac{40}{3} = \frac{30}{3} + \frac{16}{3}$
   • $2 + \frac{40}{3} = \frac{46}{3}$
   • $\frac{6}{3} + \frac{40}{3} = \frac{46}{3}$
   • $\frac{46}{3} = \frac{46}{3}$ (It checks out!)

4. Key Takeaways

• The goal is always to get the variable by itself on one side of the equals sign.
• Use inverse operations (addition/subtraction, multiplication/division) to move terms around.
• Remember the golden rule: whatever you do to one side of the equation, you must do to the other.
• Simplify each side of the equation (distribute, combine like terms) before moving terms across the equals sign.

• Always check your solution by plugging it back into the original equation.

• Common Mistakes to Avoid:

  • Forgetting to perform an operation on both sides of the equation.
  • Making sign errors when moving terms across the equals sign (e.g., $x - 5 = 10$ becomes $x = 10 - 5$ instead of $x = 10 + 5$).
  • Not simplifying fractions or combining like terms correctly.
  • Confusing terms to be added/subtracted with terms to be multiplied/divided.

5. Now Try It

Spend 15 minutes solving the following equation for $y$:

$4(2y - 3) + 7 = 3y - 5$

Success looks like you correctly isolating $y$ and finding its exact value, and then being able to plug that value back into the original equation to verify that both sides are equal.