91Ó°ÊÓ

The slope-point form is an efficient way to write the equation of a line when you know a point on the line and its slope. In this form, the equation is written as: \( y - y_1 = m (x - x_1) \). Here,

• \( (x_1, y_1) \) is a specific point on the line.
• \( m \) is the slope of the line.

If you've been given a point \((-2, -3)\) and a slope \(m = -\frac{3}{4}\), we can substitute them into the form to get the line's equation:\( y - (-3) = -\frac{3}{4}(x - (-2)) \). Notice when substituting negative values, we need to keep track of the signs properly. Simplify the equation: \( y + 3 = -\frac{3}{4} (x + 2) \) to use for further steps.

To graph a linear equation, it can be practical to use the slope-intercept form, which is \( y = mx + b \). In this form:

• \( m \) is the slope.
• \( b \) is the y-intercept (the value of y when x is 0).

Starting from the slope-point form equation \( y + 3 = -\frac{3}{4}(x + 2) \), we can expand and simplify to get the slope-intercept form. Distribute \(-\frac{3}{4} \) on the right-hand side:\( y + 3 = -\frac{3}{4}x - \frac{3}{4} \times 2 \rightarrow y + 3 = -\frac{3}{4}x - \frac{3}{2} \). Remember to subtract 3 from both sides: \( y = -\frac{3}{4}x - \frac{3}{2} - 3 \). Combine like terms to get \( y = -\frac{3}{4}x - \frac{9}{2} \). That’s your equation in slope-intercept form!

Once you have a line's equation in slope-intercept form (\( y = -\frac{3}{4}x - \frac{9}{2} \)), you can find specific points on the line by substituting values for \( x \) and solving for \( y \). Let's start with \( x = 0 \): When \( x = 0 \rightarrow y = -\frac{9}{2} \). The point \( (0, -\frac{9}{2}) \) is on the line.Now, let’s choose another \( x \) value, say 4: When \( x = 4 \rightarrow y = -\frac{3}{4}(4) - \frac{9}{2} = -3 - \frac{9}{2} \rightarrow y = -\frac{15}{2} \). So, \( (4, -\frac{15}{2}) \) is another point on the line.By substituting different \( x \) values, you can find corresponding \( y \) values and thus, more points on the line.

A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves to visualize mathematical concepts. It is made up of two perpendicular number lines: the x-axis and the y-axis.

• The horizontal line is called the x-axis.
• The vertical line is the y-axis.

Each point on the coordinate plane is identified by an ordered pair \( (x, y) \). For instance, the point \( (-2, -3) \) means \( x = -2 \) and \( y = -3 \). To plot the line from the given example, start by marking the points \((-2, -3)\), \((0, -\frac{9}{2})\), and \((4, -\frac{15}{2})\) on the coordinate plane. Then, draw a straight line passing through these points, which represents the equation \( y = -\frac{3}{4}x - \frac{9}{2} \). Understanding how to navigate the coordinate plane is crucial for graphing equations and visualizing algebraic concepts.