91Ó°ÊÓ

The slope-intercept form of a linear equation is essential for understanding and graphing lines. This form is written as \( y = mx + b \). Here, \(m\) represents the slope, and \(b\) represents the y-intercept of the line. The slope tells you how steep the line is, while the y-intercept tells you where the line crosses the y-axis.
For example, consider the equation \(2y - 3x = 0\). To convert it to the slope-intercept form, we need to solve for \(y\). Follow these steps:

• Add \(3x\) to both sides to get \(2y = 3x\).
• Divide both sides by 2 to isolate \(y\). You get \(y = \frac{3}{2}x\).

Now the equation is in the slope-intercept form \(y = mx + b \), where \(m = \frac{3}{2}\) and \(b = 0\). This means the slope is \( \frac{3}{2}\) and the y-intercept is 0.

Graphing linear equations can help you visually understand the relationship between variables. For the equation \(y = \frac{3}{2}x\), we'll plot it on the coordinate plane.
First, identify the y-intercept which is the point where the line crosses the y-axis. For this equation, the y-intercept \(b\) is 0, so start at the point (0, 0).
Next, use the slope \(m = \frac{3}{2}\) to find another point on the line. The slope \( \frac{3}{2} \) means 'rise over run,' or how much you move up for every unit moved to the right.

• From the y-intercept (0, 0), rise up 3 units (since the numerator is 3).
• Then run 2 units to the right (since the denominator is 2).

This takes you to the point (2, 3).

Draw a line through these points (0, 0) and (2, 3), extending it in both directions. This is the graph of the equation \(y = \frac{3}{2}x\).

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at a point called the origin (0, 0).
When you graph a linear equation like \(y = \frac{3}{2}x\), you place points on the coordinate plane to show the relationship between the x and y values. Each point is represented as (x, y).
For the equation \(y = \frac{3}{2}x\):

• The y-intercept is (0, 0), meaning the line starts at the origin.
• The slope tells you to move from (0, 0) to (2, 3).

By plotting these points, you create a visual representation of the equation. This helps in understanding how changes in x affect y, and vice versa.
Graphing on the coordinate plane makes abstract concepts concrete, helping you see the connections between algebra and geometry.