From Equations to Graphs
Every linear equation in two variables can be represented graphically as a straight line. The standard form of a linear equation in two variables is:
$$ax + by = c$$
Where $a$, $b$, and $c$ are constants, and $a$ and $b$ are not both zero.
The Coordinate Plane
To graph linear equations, we use the Cartesian coordinate system, consisting of:
- A horizontal x-axis
- A vertical y-axis
- The origin (0, 0) where the axes intersect
Each point on the plane is represented by an ordered pair $(x, y)$.
Slope-Intercept Form
A useful form for graphing linear equations is the slope-intercept form:
$$y = mx + b$$
Where:
- $m$ is the slope (rate of change)
- $b$ is the y-intercept (where the line crosses the y-axis)
Example: Graphing a Line
Let’s graph the equation: $y = 2x + 3$
- Identify the slope: $m = 2$
- Identify the y-intercept: $b = 3$, so the line passes through $(0, 3)$
- Use the slope to find another point: From $(0, 3)$, move right 1 unit and up 2 units to reach $(1, 5)$
- Draw the line through these points
Finding the Slope
The slope represents the steepness of a line. For two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope is calculated as:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Example: Finding the Slope
Find the slope of the line passing through points $(2, 5)$ and $(4, 9)$.
$$m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2$$
Solutions to Chapter 2 Practice Problems
- $5x + 3 = 18 \Rightarrow 5x = 15 \Rightarrow x = 3$
- $7x - 4 = 3x + 12 \Rightarrow 4x = 16 \Rightarrow x = 4$
- $2(x + 3) = 3(x - 1) + 4 \Rightarrow 2x + 6 = 3x - 3 + 4 \Rightarrow 2x + 6 = 3x + 1 \Rightarrow -x = -5 \Rightarrow x = 5$
Practice Problems
- Graph the line: $y = -3x + 2$
- Find the slope of the line passing through $(3, 7)$ and $(6, 1)$
- Convert the equation $2x + 5y = 10$ to slope-intercept form and identify the slope and y-intercept