91Ó°ÊÓ

When it comes to linear equations, understanding how to graph a line is fundamental. Every line on a graph represents a set of solutions for a given linear equation. These lines can have different slopes and intercepts, making each graph unique.

A graph of a line is straight and extends infinitely in both directions. This line can be described with equations in different forms, such as slope-intercept form or standard form. Often, to make graphing easier, we transform these equations into the slope-intercept form.

For the equation in our exercise, it simplifies to \[ y = \frac{1}{3}x. \] This indicates that the line will rise slowly as it moves from left to right due to its gentle slope.

The slope-intercept form is a special way of writing linear equations. It's given by the formula \[ y = mx + b, \] where:

• m is the slope of the line
• b is the y-intercept of the line

When dealing with the equation \[ 3x = 9y, \] we simplify it to \[ y = \frac{1}{3}x. \] Here, we've made it clear that:

• m = \frac{1}{3}
• b = 0

This makes graphing easy because we know the slope and where the line crosses the y-axis. The slope (\[ m = \frac{1}{3} \]) tells us how steep the line is: for every 3 steps horizontally, the line goes up 1 step vertically. The y-intercept (\[ b = 0 \]) means the line crosses the origin.

Linear equations have several defining features that make them predictable and easy to graph. These features help identify the behavior and shape of their graphs.

Some essential characteristics include:

• Slope (m): This number determines the slant or steepness of the line. Positive slopes rise from left to right, while negative slopes fall.
• Y-Intercept (b): This is where the line crosses the y-axis. A y-intercept of 0 means it passes through the origin.
• Consistency: Linear equations produce straight lines with no curves. Each input (x) has exactly one corresponding output (y).

For the equation in our exercise \[ 3x = 9y, \] converting it to \[ y = \frac{1}{3}x \] allowed us to see its key characteristics: a slope (\[ m = \frac{1}{3} \]) and a y-intercept (\[ b = 0 \]). These traits mean our graph will be a straight, diagonal line passing through the origin and rising slowly.