91Ó°ÊÓ

The point-slope formula is a helpful tool for writing the equation of a line when you know a point on the line and its slope. This formula is written as \(y - y_1 = m(x - x_1)\), where \( (x_1, y_1) \) is the given point and \( m \) is the slope.

Using the point-slope formula allows you to directly plug in the known quantities to find the linear equation quickly. For instance, in the exercise where the point is \( (0, -5) \) and the slope \(-2\), substituting these values into the point-slope formula yields: \((y + 5) = -2(x - 0))\).

The next step is to simplify this equation to make it easier to understand and use for graphing purposes.

The coordinate plane is a two-dimensional surface where points are determined by pairs of numbers called coordinates. These coordinates are written as \( (x, y) \), where \( x \) represents the horizontal position, and \( y \) represents the vertical position.

For example, the point \( (0, -5) \) means the location is on the y-axis 5 units below the origin. This point serves as a starting position for graphing our line.

Once we have plotted \( (0, -5) \) on the coordinate plane, we can use the slope—to direct us on how the line continues. The slope \( m \) of the line tells us how to move from one point to the next. An important note is:

• If the slope is negative, we move downward and to the right.
• If the slope is positive, we move upward and to the right.

In our case, a slope of \(-2\) means moving 1 unit to the right and 2 units downwards.

The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line. It is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

After using the point-slope formula and simplifying, we ended up with the equation \( y = -2x - 5 \). This is in slope-intercept form. Here:

• The slope \(-2\) tells us the rate of change in \( y \) for each unit change in \( x \).
• The y-intercept \(-5\) represents the point where the line crosses the y-axis.

To graph this, we start with the y-intercept \(-5\), plot the point then use the slope to find more points. In our case, we move 1 unit to the right and 2 units down to find subsequent points. Connect these points to reveal the line.

Using the slope-intercept form makes it easy to visualize and understand the graph of a linear equation.