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The coordinate plane is a fundamental tool in math used to graph equations, including linear equations. Picture a flat surface marked by two intersecting number lines: the horizontal axis, known as the x-axis, and the vertical axis, called the y-axis. These axes divide the plane into four regions called quadrants, labeled I through IV.

• The x-axis runs horizontally and values increase as you move to the right and decrease as you move to the left.
• The y-axis runs vertically with values increasing as you move up and decreasing as you move down.

Each point on a coordinate plane can be identified using an ordered pair \( (x, y) \), where \( x \) is the horizontal position and \( y \) is the vertical position relative to the origin \( (0, 0) \), which is the point where both axes meet. Understanding how to navigate and plot points on this plane is essential for graphing linear equations effectively.

A linear equation creates a straight line when plotted on a coordinate plane. It is an equation of the form \( y = mx + b \), commonly known as the slope-intercept form. This form is particularly handy because it immediately informs us about two key features of the line:

• The slope \( m \) tells us how steep the line is. It shows the rate at which the y-value changes for every unit increase in x.
• The y-intercept \( b \) is the point where the line crosses the y-axis. When x is zero, y is equal to b.

Linear equations are called 'linear' because their graph is always a straight line. The equation \( y = \frac{2}{3}x - 1 \) means that for every 3 units you move to the right on the x-axis, you move 2 units up on the y-axis from the y-intercept -1. Knowing this makes it simple to find additional points on the line.

Graphing lines on the coordinate plane involves plotting points and connecting them to reveal the straight line described by a linear equation.

• Begin with the y-intercept, which is a starting point, such as \( (0, -1) \) in the given exercise.
• Use the slope to determine the next point. From \( (0, -1) \), a slope of \( \frac{2}{3} \) directs you to move 2 units up and 3 units right, landing on another point like \( (3, 1) \).

After points are plotted, draw a line through them that extends indefinitely in both directions. Check your work by ensuring the line aligns with the slope. Graphing reveals the relationship and behavior of the equation visually, making it easier to understand and analyze.