91Ó°ÊÓ

The slope is a fundamental concept in algebra when dealing with lines on a graph. Essentially, the slope measures the steepness or incline of a line. More formally, it shows how much the line rises or falls for each unit of run. In our given exercise, the slope is \( m = \frac{2}{3} \). This means that for every 3 units we move horizontally (to the right on a graph), the line moves 2 units vertically (upwards).
Understanding this ratio helps in visualizing how quickly a line rises or drops as you move along it.

• Rise: The vertical change between two points on the line (in this case, 2).
• Run: The horizontal change between those same two points (in this case, 3).

This concept is not only essential for graphing lines but also for solving problems involving linear equations and predicting future points on the line.

The point-slope form is a convenient way to write the equation of a line when you know one point on the line and the slope. The general formula is \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) represents a known point on the line, and \( m \) is the slope we talked about earlier.
In the exercise, the given point is \((3, 2)\), and the slope \( m = \frac{2}{3} \).

• Step 1: Substitute \( x_1 = 3 \), \( y_1 = 2 \), and \( m = \frac{2}{3} \) into the formula to create the equation of our line.
• Step 2: This gives us \( y - 2 = \frac{2}{3}(x - 3) \).

Using the point-slope form, you can quickly determine other points on the line by selecting different \( x \) values and solving for \( y \). This method ensures that all calculated points will lie on the same straight line.

Once we have a point-slope equation, it's possible to simplify it to the slope-intercept form \( y = mx + b \) for ease of further computation or graphing. For our exercise, the step-by-step solution guides us to simplify by distributing and combining like terms.

• Expand: Distribute \( \frac{2}{3} \) across \( (x - 3) \) to simplify the right side of \( y - 2 = \frac{2}{3}(x - 3) \).
• Simplify: Solve for \( y \) by adding 2 to both sides, yielding \( y = \frac{2}{3}x \).

This is the slope-intercept form, which makes it straightforward to choose values for \( x \) to find corresponding \( y \). By plugging in different \( x \) values, such as 6, 9, or 0, you can glean additional points like \((6, 4)\), \((9, 6)\), and \((0, 0)\) all accurately lying on the line represented by our equation. This form is especially useful for quickly determining the y-value of any point on the line once an x-value is given.