A linear equation describes a straight line. It's an equation where the variables, usually $x$ and $y$, aren't raised to any power greater than one. You won't find any $x^2$ or $\sqrt{y}$ here. The relationship between the variables is constant and straightforward, which is why it graphs as a perfectly straight line.

One common way to write a linear equation is the standard form:

In this form, $x$ and $y$ are our variables. $A$, $B$, and $C$ are constants, which are just fixed numbers. For example, $2x + 3y = 6$ is a linear equation in standard form. Here, $A=2$, $B=3$, and $C=6$. This form is neat and organized, but it isn't always the most intuitive for understanding the line's properties at a glance.

A more user-friendly format is the slope-intercept form, which is probably the one you'll see most often.

This form tells you two key things about the line just by looking at it: its slope ($m$) and its y-intercept ($b$).

• The slope ($m$) measures the steepness of the line. It's often described as "rise over run." A slope of 2 means that for every 1 unit you move to the right on the graph, you move 2 units up. A negative slope means the line goes down as you move from left to right.

• The y-intercept ($b$) is the point where the line crosses the vertical y-axis. It's the value of $y$ when $x$ is 0.

Intercepts are the points where a line crosses the axes. While the y-intercept is obvious in slope-intercept form, we need a reliable way to find both intercepts for any linear equation. The rule is simple.

• To find the y-intercept, set $x = 0$ and solve for $y$.
• To find the x-intercept (where the line crosses the horizontal x-axis), set $y = 0$ and solve for $x$.

Let's use our earlier example, $2x + 3y = 6$.

First, find the y-intercept by setting $x=0$:

The y-intercept is at the point $(0, 2)$.

Now, find the x-intercept by setting $y=0$:

The x-intercept is at the point $(3, 0)$. These two points are often all you need to draw the entire line.

You can easily convert from standard form to slope-intercept form by just isolating $y$. For $2x + 3y = 6$:

Now we can see the slope is $-\frac{2}{3}$ and the y-intercept is 2, which matches what we found. Every form of the equation tells the same story about the line, just in a different way.

Let's check what you've learned.

Understanding these basics is the first step toward working with more complex systems and real-world problems.