Problem 34 - Algebra 1: Concepts and Skills

Question

Graph the equation. $$y=\frac{2}{3} x$$

Step-by-Step Solution

Answer

The graph of the equation \(y = \frac{2}{3}x\) is a straight line that passes through the origin (0,0) and the point (3,2). The line slopes upward, indicating a positive slope of \(\frac{2}{3}\).

Step 1: Understand the Slope

The slope of the line (\(m\)) is \(\frac{2}{3}\). This means for every 3 units you move to the right along the x-axis (the horizontal axis), you must move 2 units up along the y-axis (the vertical axis).

Step 2: Identify and Mark the Y-Intercept

Here, the y-intercept (b) is 0 which is where the line crosses the y-axis. So mark this point \(0, 0\) on y-axis.

Step 3: Use the Slope to Find a Second Point

To utilize the slope, from the y-intercept, move 3 units to the right along the x-axis and then two units up along the y-axis (as the slope is \(\frac{2}{3}\)). This gives you a second point on the graph at coordinates \((3,2)\). Mark this point on the graph.

Step 4: Draw the Line

Draw a straight line through the points marked in Step 2 and Step 3. This line represents the graph of the equation \(y = \frac{2}{3}x\).

Key Concepts

Slope

When graphing linear equations, understanding the concept of slope is a crucial step. The slope is often symbolized by the letter \(m\), and it represents the steepness or incline of a line on the coordinate plane. In the equation \(y = \frac{2}{3}x\), the slope is given as \(\frac{2}{3}\). This fractional expression tells us that for every 3 units moved horizontally to the right, the line moves 2 units vertically upwards.

Here's how to interpret this:

• The numerator of the fraction, 2, indicates the rise or the change in the y-values as you move along the line.
• The denominator, 3, indicates the run or horizontal movement across the x-values.

Recognizing this pattern helps you plot additional points on the plane, which is essential in drawing the graph of the linear equation. Just pick any horizontal distance (run) and apply the slope ratio, so you can find the corresponding vertical movement (rise). This is the essence of how slope helps us to understand and create linear graphs.

Y-Intercept

The y-intercept is another fundamental concept in graphing linear equations. It represents the point where the line crosses the y-axis. In an equation of the form \(y = mx + b\), \(b\) is the y-intercept. For the equation \(y = \frac{2}{3}x\), there is no constant term, which means the y-intercept is 0.

Here are a few key points about y-intercepts:

• The y-intercept is always located at a point where \(x = 0\).
• In our example, the y-intercept is the origin point \((0,0)\), where both x and y coordinates are zero.
• This starting point is crucial because it provides a firm location to begin using the slope to find other points.

Visualizing the y-intercept as the starting anchor allows us to understand the placement and direction of the line on the plane. Without it, placing the line correctly would be tricky.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graphically represent equations. It comprises two axes: the horizontal x-axis and the vertical y-axis. When graphing linear equations like \(y = \frac{2}{3}x\), understanding this plane is vital to correctly plotting points and drawing the line those points form.

Important features of the coordinate plane include:

• It is divided into four quadrants by the intersection of the x-axis and y-axis.
• Each point on the plane is represented by a pair of numbers \((x, y)\), known as coordinates.

The coordinate plane allows you to graph the line by first plotting the y-intercept location. Then, using the slope to navigate through points across the plane, you connect these points to visualize the linear relationship. This spatial representation is pivotal in understanding how equations translate into visual graphs, helping us better interpret mathematical relationships.