91Ó°ÊÓ

The point-slope form of a linear equation is crucial for understanding how to graph lines using specific points and slopes. The general formula is:
y - y_1 = m(x - x_1)
Here,

• y - y_1 represents the difference in the y-coordinates between any point on the line and a specific point (x_1, y_1) on the line
• m is the slope of the line
• x - x_1 represents the difference in the x-coordinates between any point on the line and the specific point (x_1, y_1) on the line

To solve the given problem, you substitute the coordinates of the given point and the slope into the equation. For example, with point (-3, 3) and slope 2:
y - 3 = 2(x + 3). This forms the foundation to move towards an equation that can reveal more about the line, such as the slope-intercept form.

The slope-intercept form makes graphing more straightforward. The general equation is:
y = mx + b
Where:

• m represents the slope
• b is the y-intercept where the line crosses the y-axis

Continuing from our point-slope form equation, y - 3 = 2(x + 3), we first distribute the slope and combine like terms:
y - 3 = 2x + 6
We then solve for y to convert to the slope-intercept form:
y = 2x + 9
In this form, it's clear that the slope (m) is 2 and the y-intercept (b) is 9. This means the line rises 2 units for every horizontal move of 1 unit. It also tells us where to begin plotting the line on the graph, starting at the y-axis at point (0, 9).

Plotting points using both the y-intercept and the slope makes graphing linear equations precise and accurate. Here's how you can graph the solution:
First, identify the y-intercept from the slope-intercept form equation, y = 2x + 9. The y-intercept is 9, so plot the point (0, 9) on the graph.
Next, use the slope to find another. Since the slope is 2, this signifies a 'rise over run' of 2/1:

• Starting from (0, 9) move up 2 units to 11
• Then, move 1 unit to the right to reach (1, 11)

Draw a point at (1, 11). Now that you have two points, (0, 9) and (1, 11), draw a straight line through them, extending it in both directions. This line represents the linear equation y = 2x + 9.
Graphing points this way ensures accuracy and helps visualize the relationship dictated by the linear equation.