91Ó°ÊÓ

The slope-intercept form is one of the most common ways to write a linear equation, making it straightforward to visualize the line's behavior on a graph. Expressed as \( y = mx + c \), this form reveals two crucial pieces of information: the slope \( m \) and the y-intercept \( c \). The slope \( m \) tells us how steep the line is and in which direction it tilts. In our example, with the equation \( y = -\frac{1}{2}x + 3 \), the slope is \(-\frac{1}{2}\), meaning the line decreases in height as \( x \) increases. The y-intercept \( c \) is where the line crosses the y-axis, which in this case is 3. This means that when \( x = 0 \), the value of \( y \) is directly the y-intercept, giving us the point \((0, 3)\) on the graph.

Understanding intercepts is key to effectively graphing linear equations. An intercept is a point where the line crosses an axis on a coordinate plane. There are two types of intercepts:

• Y-Intercept: This is the value of \( y \) where the line meets the y-axis. Since the x-coordinate is zero, in our case, the y-intercept is \( (0, 3) \).
• X-Intercept: This is the value of \( x \) when the line intersects the x-axis, making \( y = 0 \). By substituting \( y = 0 \) into the equation, we solved \( \frac{1}{2}x = 3 \) to determine that \( x = 6 \), giving us the x-intercept \( (6, 0) \).

These intercepts provide strategic points that guide the drawing of the complete line for the equation \( \frac{1}{2}x + y = 3 \). After finding these intercepts, plot them, and then draw a line through them, which should extend in both directions.

The coordinate plane is an essential tool for graphing equations and understanding their geometric representation. It consists of two perpendicular axes: the x-axis, which runs horizontally, and the y-axis, which runs vertically. They intersect at a point called the origin, designated as \( (0, 0) \). Each point on this plane is identified by a pair of coordinates \((x, y)\). For the equation \( \frac{1}{2}x + y = 3 \), after converting to slope-intercept form, you can graph it by using its x- and y-intercepts as reference points. Start by marking the y-intercept \( (0, 3) \) and the x-intercept \( (6, 0) \). Connect these points with a straight line. This plotted line not only shows the solutions to the equation but visually represents the relationship between x and y throughout infinite points on that line.