Problem 38 - Introductory Algebra for College Students

Question

Graph each linear equation using the slope and y-intercept. $$y=-\frac{4}{3} x$$

Step-by-Step Solution

Answer

The graph of the equation \(y = -\frac{4}{3}x\) is a straight line through the origin with slope -4/3, moving four steps down for every three steps to the right.

Step 1: Identify the y-intercept

The equation \(y= -\frac{4}{3}x\) is the same as \(y=-\frac{4}{3}x+0\). Thus, the y-intercept (where the line crosses the y-axis) is at the origin (0,0).

Step 2: Understand the Slope

The slope of this line is -4/3. This means for each three units you go to the right on x-axis, you go four units down on the y-axis, since it's negative.

Step 3: Graph the Line

Plot the y-intercept at (0,0). From there, follow the slope by moving three units to the right and four units down repeatedly until a line can be drawn through the plotted points, clearly representing the linear equation.

Key Concepts

Slope

The slope is a key feature of linear equations, expressed as the ratio of how much the line goes up or down for each step it goes right. It tells us how steep the line is. In the equation \(y = -\frac{4}{3}x\), the slope is \(-\frac{4}{3}\). This means:

• For every 3 units you move to the right, the line drops 4 units.
• The negative sign indicates the line goes downward as you move from left to right.

Using the slope, you can easily plot points along the line once you know where it starts. Remember, the slope is often represented as \(\frac{rise}{run}\), where 'rise' is how much you go up or down, and 'run' is how much you move left or right. In this case, it’s a fall of 4 (negative rise) per 3 right units of run. This information is crucial for correctly graphing the line.

y-intercept

The y-intercept is where the line crosses the y-axis. It shows the value of \(y\) when \(x = 0\). In our equation \(y = -\frac{4}{3}x + 0\), the y-intercept is 0. This means the line crosses the y-axis at the point (0,0).
The y-intercept provides a starting point on the graph, which is essential for drawing the line:

• Begin by plotting the y-intercept.
• Use this point as the anchor to apply the slope.

By understanding the y-intercept, you can more accurately represent linear equations on a graph. It simplifies the graphing process, as you always know where to begin drawing the line.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph equations. It consists of two axes:

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

These axes split the plane into four quadrants. Each point on this plane is identified by a pair of numbers (x, y), called coordinates.
To graph a linear equation like \(y = -\frac{4}{3}x\), follow these steps:

• Start at the y-intercept.
• Move according to the slope.

Understanding the coordinate plane helps in visualizing how the slope and y-intercept interact. It's a foundational tool for graphing any linear equation and is used to identify the placement and direction of lines. Mastering the coordinate plane allows you to see math come to life visually.