91Ó°ÊÓ

In order to understand the behavior of a line, we first need to determine its slope. The slope, often represented by the letter \(m\), measures how steep a line is. It's calculated by dividing the change in the \(y\)-values by the change in the \(x\)-values between two points. This formula can be expressed as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This fraction is called the "rise over run."

• "Rise" is the difference in the \(y\)-values \((y_2 - y_1)\).
• "Run" is the difference in the \(x\)-values \((x_2 - x_1)\).

For the points (-5, -5) and (10, 7), the slope calculation looks like this: \[ m = \frac{7 - (-5)}{10 - (-5)} = \frac{12}{15} = \frac{4}{5} \]. This result tells us that for every 5 units we move horizontally, the line rises 4 units vertically. A positive slope like \(\frac{4}{5}\) means the line ascends from left to right.

Once we've calculated the slope, we can form an equation of the line using the point-slope form. This formula is particularly useful when you know one point on the line and the slope. It’s expressed as: \[ y - y_1 = m(x - x_1) \].
Here, \((x_1, y_1)\) is a point on the line and \(m\) is the slope we calculated. By using one of our points, say (-5, -5), along with the slope \(\frac{4}{5}\), we can plug them into the point-slope formula: \[ y - (-5) = \frac{4}{5}(x - (-5)) \].
This equation represents the line but isn't in a format commonly used for graphing or further calculations. It provides a simple method to form a line equation quickly with minimal information.

To make the line equation more accessible, we convert it to the slope-intercept form, which is \(y = mx + b\). In this form, \(m\) is the slope, and \(b\) is the \(y\)-intercept, the point where the line crosses the \(y\)-axis.
Starting with the point-slope equation \[ y + 5 = \frac{4}{5}(x + 5) \], expand and simplify it:

• Distribute \(\frac{4}{5}\) to both \(x + 5\): \(y + 5 = \frac{4}{5}x + \frac{4}{5} \times 5\).
• Calculate \(\frac{4}{5} \times 5\) to get \(4\): \(y + 5 = \frac{4}{5}x + 4\).
• Subtract 5 from both sides to solve for \(y\): \(y = \frac{4}{5}x - 1\).

This results in the equation \(y = \frac{4}{5}x - 1\), where \(-1\) is the \(y\)-intercept. This format is useful for graphing as it directly shows the slope and where the line hits the \(y\)-axis. Verifying this with the point (10, 7) confirms the equation, as plugging \(x = 10\) results in \(y = 7\).