Solving Linear Equations: A Step-by-Step Guide

Welcome, Mathletes! Today's lesson focuses on mastering linear equations. We'll explore the fundamental principles, practice with examples, and set you up for success with targeted homework. Let's get started!

Key Concepts

A linear equation is an equation in which the highest power of the variable is 1. Our goal is to isolate the variable on one side of the equation to find its value. Remember, whatever operation you perform on one side of the equation, you MUST perform on the other side to maintain equality. We can use the following steps to solve linear equations:

  1. Remove Parentheses: If the equation contains parentheses, use the distributive property to expand and eliminate them. For example: $3(2x - 5) = 15$ becomes $6x - 15 = 15$.
  2. Combine Like Terms: Simplify each side of the equation by combining like terms.
  3. Move Variables to One Side: Use addition or subtraction to gather all terms containing the variable on one side of the equation.
  4. Remove Addition/Subtraction: Use addition or subtraction to isolate the term containing the variable.
  5. Remove Multiplication/Division: Use multiplication or division to solve for the variable.

Examples

Let's work through a few examples to illustrate these steps:

Example 1: Solve for $x$ in the equation $3x + 5 = 12$.

  1. Subtract 5 from both sides: $3x + 5 - 5 = 12 - 5$, which simplifies to $3x = 7$.
  2. Divide both sides by 3: $\frac{3x}{3} = \frac{7}{3}$, which gives us $x = \frac{7}{3}$.

Example 2: Solve for $x$ in the equation $\frac{3}{7}x + 9 = 15$.

  1. Subtract 9 from both sides: $\frac{3}{7}x = 6$.
  2. Multiply both sides by $\frac{7}{3}$: $x = 6 * \frac{7}{3} = 14$.

Example 3: Solve for $x$ in the equation $6(x + 4) - 2x + 7 = 3x - 12$.

  1. Distribute: $6x + 24 - 2x + 7 = 3x - 12$
  2. Combine Like Terms: $4x + 31 = 3x - 12$
  3. Subtract 3x from both sides: $x + 31 = -12$
  4. Subtract 31 from both sides: $x = -43$

Example 4: Solve for $x$ in the equation $(\frac{1}{3}x + \frac{1}{4} = x - \frac{1}{6}) * 12$

  1. Distribute: $4x + 3 = 12x - 2$
  2. Subtract 4x from both sides: $3 = 8x - 2$
  3. Add 2 to both sides: $5 = 8x$
  4. Divide both sides by 8: $\frac{5}{8} = x$

Homework Assignment

For practice, please complete the following problems from your textbook:

  • Pages 22-23: Problems #23-40

Remember to show your work and check your answers! Good luck, and don't hesitate to ask questions in class or during office hours.