Linear equations — 7-topic bundle

The Cartesian coordinate system, also known as the rectangular coordinate system, is a way to uniquely identify every point on a plane using two numbers.

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When exploring linear relationships, you'll mainly use rules, tables, and graphs. A rule is an equation that describes the relationship between two or more variables. Often, for two variables, you'll see y as the subject of the equation.

A relationship between variables x and y can be shown as a table or as a graph. Points on a graph are ordered pairs (x, y) where x is the horizontal unit from the origin and y is the vertical unit from the origin.

Linear relationships, as you've seen, often describe how two variables, like x and y, are connected by a rule (an equation). When graphed, these relationships form straight lines. Two key features help us understand and sketch these lines: the intercepts and the gradient.

The x-intercept is the point where your straight line crosses the x-axis. At this point, the y-coordinate is always 0.
To find it, you simply set y = 0 in your linear equation and solve for x.

A rule is an equation that shows how two or more variables are connected. For linear relationships, this rule will give you a straight line when graphed.

Think of these three as different ways to show the same relationship:

• Rule (Equation): This is the mathematical formula, like y = 2x - 1.
• Table of Values: This lists several x values and their corresponding y values based on the rule.
• Graph: This is a visual representation of the (x, y) pairs plotted on a coordinate plane, forming a line for linear rules.

You've already covered the Cartesian plane, which is your map, and how to plot points using coordinates (x, y). Remember, x tells you how far horizontally from the origin (0,0) to go, and y tells you how far vertically. The plane is divided into four quadrants, with the first quadrant having positive coordinates for both x and y.