Linear equations are the building blocks of algebra, describing straight-line relationships between variables. They're essential for modeling real-world scenarios, from economics to physics. Understanding their components and how to graph them unlocks powerful predictive abilities.

These equations have a constant rate of change, represented by the slope. By mastering linear equations, you'll be able to make predictions, analyze trends, and solve problems across various fields. It's a fundamental skill that sets the stage for more complex mathematical concepts.

Linear Equations

Components of linear equations

• $y = a + bx$ represents the general form of a linear equation
  • $y$ represents the dependent variable, the variable being predicted or explained
  • $x$ represents the independent variable, the variable doing the predicting or explaining
  • $a$ represents the y-intercept, the value of $y$ when $x = 0$ (the point where the line crosses the y-axis)
  • $b$ represents the slope, the change in $y$ divided by the change in $x$ (steepness and direction of the line)
    • Slope is calculated as $\frac{\Delta y}{\Delta x}$ or $\frac{y_2 - y_1}{x_2 - x_1}$
      • $\Delta y$ represents the vertical change between two points
      • $\Delta x$ represents the horizontal change between two points
    • A positive slope indicates an increasing line (moving up from left to right)
    • A negative slope indicates a decreasing line (moving down from left to right)
    • A slope of 0 indicates a horizontal line (no change in $y$ as $x$ changes)
    • An undefined slope indicates a vertical line (no change in $x$ as $y$ changes)
• Point-slope form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope

Predictions using linear equations

• To predict $y$ values for given $x$ values, substitute the $x$ value into the linear equation and solve for $y$
  • Follow the order of operations (PEMDAS) when solving: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Example: For the linear equation $y = 3x - 2$, predict the value of $y$ when $x = 5$
  1. Substitute $x = 5$ into the equation: $y = 3(5) - 2$
  2. Multiply: $y = 15 - 2$
  3. Subtract: $y = 13$
  • The predicted value of $y$ when $x = 5$ is 13
• Function notation: $f(x) = 3x - 2$ is equivalent to $y = 3x - 2$, allowing for more concise representation of functions

Graphing of linear equations

• To graph a linear equation, follow these steps:
  1. Identify and plot the y-intercept $(0, a)$ on the coordinate plane
  2. Calculate the slope $b$ using the given equation or two points on the line
  3. Use the slope to find another point on the line by moving $b$ units vertically and 1 unit horizontally from the y-intercept (rise over run)
  4. Plot the second point on the coordinate plane
  5. Connect the two points with a straight line extending infinitely in both directions
• Characteristics of a linear equation graph include:
  • A straight line, indicating a constant rate of change (slope) between any two points
  • Extends infinitely in both the positive and negative directions
  • Crosses the y-axis at the y-intercept $(0, a)$
• Example: Graph the linear equation $y = -\frac{1}{2}x + 3$
  • The y-intercept is $(0, 3)$
  • The slope is $-\frac{1}{2}$, meaning the line decreases by 1 unit vertically for every 2 units moved horizontally
  • Plot the y-intercept and use the slope to find another point, such as $(-2, 4)$ or $(2, 2)$
  • Connect the points with a straight line

Relationships between linear equations

• Parallel lines have the same slope but different y-intercepts
• Perpendicular lines have slopes that are negative reciprocals of each other
• A system of linear equations consists of two or more linear equations that are considered simultaneously