The document you've uploaded contains multiple math problems and solutions from your math course. I'll go through the key principles and techniques for solving the problems presented. I'll also break down the steps for each problem to ensure you understand the methods. Here’s a breakdown:

Problem Breakdown and Techniques

1. Linear vs. Nonlinear Equations

• Linear equations are equations that graph as straight lines. They have the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• Nonlinear equations don’t graph as straight lines and include terms like $x^2$, square roots, etc.

For example:

• Equation 1: $2x + 6 = 6x - 6$: This is linear because it's of the form $ax + b = cx + d$.
• Equation 2: $x(x - 4) - 9 = 2 + 4$: This is nonlinear because it includes a product of $x$'s.

2. Solving Linear Equations

• The goal is to isolate the variable.
• Example 1: Solve $5(3 - 5x) + 3 = 5x$:
  • Distribute: $15 - 25x + 3 = 5x$
  • Simplify: $18 - 25x = 5x$
  • Combine like terms: $18 = 30x$
  • Solve for $x$: $x = \frac{18}{30} = \frac{3}{5}$

3. Solving Using Substitution or Graphing

• For systems of equations, you can use substitution or graphing.
• Example: Graphing $y = x + 3$ and $y = -x + 2$, the intersection point is the solution.

4. Checking Solutions

• Substitute the value of $x$ back into the original equation to verify it holds true.

5. Finding Intercepts (x and y)

• x-intercept: Set $y = 0$ and solve for $x$.
• y-intercept: Set $x = 0$ and solve for $y$.

Example:

• For the equation $2x - 5y = -10$:
  • x-intercept: Set $y = 0$, solve $2x = -10 \Rightarrow x = -5$.
  • y-intercept: Set $x = 0$, solve $-5y = -10 \Rightarrow y = 2$.

6. Graphing Linear Equations

• Plot the intercepts and draw a straight line through them. If only one intercept is available, pick another point on the line to complete the graph.

7. Using Slope-Intercept Form

• The slope-intercept form is $y = mx + b$, where:
  • $m$ is the slope: $m = \frac{\text{change in } y}{\text{change in } x}$
  • $b$ is the y-intercept.

Example: Find the equation of a line passing through the points $(1, 5)$ and $(3, 6)$:

• Find slope: $m = \frac{6-5}{3-1} = \frac{1}{2}$.
• Use the point $(1, 5)$ and the slope to write the equation: $y - 5 = \frac{1}{2}(x - 1)$.

Shortcuts and Techniques:

1. Solving Linear Equations Quickly:

   • Always aim to simplify the equation step-by-step: eliminate parentheses, combine like terms, and move variables to one side.

2. Finding the Slope from Two Points:

   • Use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ to quickly find the slope between two points. This is key for writing the equation of a line.

3. Intercepts:

   • To find the x-intercept, always set $y = 0$.
   • To find the y-intercept, set $x = 0$.

Do you want a more detailed explanation of any of these techniques, or do you have specific questions about a particular problem?

Here are some related questions that might help:

1. How do I graph a linear equation using slope and intercepts?
2. How do I solve for $x$ when I have a fractional equation?
3. Can you explain the process of solving systems of equations by substitution or elimination?
4. How do I determine if a graph represents a linear equation?
5. What is the difference between direct variation and linear equations?

Tip: Always check your work by plugging the solution back into the original equation. This ensures that your answer is correct!