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The point-slope form is a straightforward method to write the equation of a line when you know a point on the line and its slope. The formula is given by \[ y - y_1 = m(x - x_1) \]where:

• \(y_1\) and \(x_1\) are the coordinates of the known point on the line,
• \(m\) is the slope of the line.

This form is especially useful because it allows you to easily write an equation when given specific data. In our exercise, we plugged in the point (-2, -5) and slope \(\frac{3}{4}\) directly into the equation:\[ y + 5 = \frac{3}{4}(x + 2) \]Notice how it involves simple substitution! Once the values are inserted, you can simplify further, if needed, to put it into a different form—like the slope-intercept form! This transition helps highlight the relationship between point-slope form and other types of linear equations.

The slope-intercept form of a line gives a clear view of the slope and the y-intercept directly in the equation. It is expressed as:\[ y = mx + b \]Here:

• \(m\) is the slope, indicating how steep the line is,
• \(b\) is the y-intercept, showing where the line crosses the y-axis.

This form is particularly handy when you need to graph a line quickly. In our exercise, after using the point-slope form, we simplify the equation to get to this form:\[ y = \frac{3}{4}x + \frac{3}{2} - 5 \]Simplifying further, we obtain:\[ y = 0.75x - 3.5 \]This equation tells us directly that for every increase of 1 in \(x\), \(y\) increases by 0.75, and the line crosses the y-axis at -3.5. Understanding this form can make solving real-world problems with linear equations much simpler.

Graphing a line is an essential skill in mathematics, allowing you to visualize relations and changes between variables. To plot our line, we start with the slope-intercept form obtained:\[ y = 0.75x - 3.5 \]Here's how you can graph it step by step:

• Identify the y-intercept. In this case, it's -3.5, which is where the line will cross the y-axis.
• Use the slope (0.75), which can be interpreted as 'rise over run.' For every 4 units (denominator of \(\frac{3}{4}\)) you move horizontally (x-axis), you move 3 units vertically (y-axis).
• Start at the y-intercept (-3.5) and use the slope to find another point. From (-3.5), move up 0.75 units for every 1 unit you move right.
• Plot these points and draw a straight line through them.

This method ensures accuracy and helps you see exactly how the line behaves. Remember, a graph provides a visual representation of a function, making abstract concepts more tangible and easier to understand.