11.1 Use the Rectangular Coordinate System

Rectangular Coordinate System

Points on coordinate systems

The rectangular coordinate system (also called the Cartesian coordinate system) is a two-dimensional plane formed by two number lines crossing at right angles. The horizontal line is the x-axis, and the vertical line is the y-axis. The point where they intersect is called the origin, with coordinates (0, 0).

Every point on this plane is described by an ordered pair $(x, y)$:

• The x-coordinate tells you how far left or right the point is from the origin
• The y-coordinate tells you how far up or down the point is from the origin

To plot a point like (3, −2):

1. Start at the origin
2. Move 3 units to the right (positive x means right)
3. Move 2 units down (negative y means down)
4. Mark the point

The coordinate plane is divided into four quadrants, numbered counterclockwise starting from the upper right:

• Quadrant I: $x > 0$ and $y > 0$ (upper right)
• Quadrant II: $x < 0$ and $y > 0$ (upper left)
• Quadrant III: $x < 0$ and $y < 0$ (lower left)
• Quadrant IV: $x > 0$ and $y < 0$ (lower right)

Points that sit directly on an axis (like (0, 5) or (−3, 0)) aren't in any quadrant.

Tables for linear equations

A linear equation in two variables has the form $y = mx + b$, where $m$ is the slope (steepness of the line) and $b$ is the y-intercept (where the line crosses the y-axis). Its graph is always a straight line.

To build a table of solutions, pick a few x-values and plug them into the equation to find the matching y-values. For example, with $y = 2x + 1$:

Each row gives you a point you can plot. Connect the points, and you've got the graph of the equation. Picking at least three x-values is a good habit because if one point is off, you'll notice it doesn't line up with the others.

Solutions using graphs

The graph of an equation shows every point $(x, y)$ that makes the equation true. That means:

• Any point on the line is a solution to the equation
• Any point off the line is not a solution

To verify whether a specific point is a solution, substitute its x and y values into the equation. For $y = 2x + 1$, check whether (3, 7) is a solution: plug in $x = 3$ and see if $y = 7$. Since $2(3) + 1 = 7$, yes, it checks out.

To find solutions from a graph:

1. Plot the equation on the coordinate plane
2. Identify any point that lies on the graphed line
3. Read off its x- and y-coordinates to get the solution pair

X-coordinates vs y-coordinates

The x-coordinate (formally called the abscissa) always comes first in an ordered pair, and the y-coordinate (the ordinate) always comes second. The order matters: (3, 5) and (5, 3) are two completely different points.

On the coordinate plane, x-values increase as you move right and decrease as you move left. Y-values increase as you move up and decrease as you move down.

Ordered pairs in real-world contexts

Ordered pairs show up whenever you need to connect two pieces of information:

• Locations: On a city grid, (3, 4) could mean 3 blocks east and 4 blocks north of a starting point
• Data relationships: A company's sales record might use (2, 5) to mean 2 years after founding and $5 million in total sales
• Science: A temperature reading at a specific time, like (3, 72), could mean 72°F at 3:00 PM

In each case, the first number and second number represent different quantities, so keeping them in the right order is essential.

Coordinate Geometry

Coordinate geometry (also called analytic geometry) combines algebra with geometry by placing shapes on the coordinate plane. This lets you use equations and formulas to calculate distances, midpoints, and slopes rather than relying only on diagrams. You'll use these skills heavily as you move into more advanced graphing topics.