Problem 8 - Elementary Algebra

Question

For the following problems, graph the equations. $$ 3 x-2 y=6 $$

Step-by-Step Solution

Answer

Question: Graph the equation \(3x - 2y = 6\) and identify the slope and y-intercept. Answer: The graph of the equation \(3x - 2y = 6\) is a straight line that passes through the points (0, -3) and (2, 0). The slope of the line is \(\frac{3}{2}\) and the y-intercept is -3.

Step 1: Rewrite the equation in slope-intercept form

To rewrite the equation \(3x - 2y = 6\) in slope-intercept form, solve for y: $$ -2y = -3x + 6 $$ $$ y = \frac{3}{2}x - 3 $$ Now the equation is in slope-intercept form: \(y = \frac{3}{2}x - 3\)

Step 2: Identify the slope and y-intercept

The slope(m) and y-intercept(b) can be identified from the equation: \(y = \frac{3}{2}x - 3\) Slope (m): \(\frac{3}{2}\) Y-intercept (b): -3

Step 3: Plot the y-intercept on graph

The y-intercept is the point where the line crosses the y-axis. In this case, it is -3. So, plot a point at (0, -3).

Step 4: Plot additional points using the slope

The slope \(\frac{3}{2}\) is equal to the "rise" divided by the "run". It tells us that for an increase of 2 units in x (moving to the right on the x-axis), there's an increase of 3 units in y (moving upward on the y-axis). Starting from the y-intercept point (0, -3), move 2 units to the right and 3 units up. This will give you a point at (2, 0). Plot this point on the graph.

Step 5: Draw the line

With the two plotted points (0, -3) and (2, 0), draw a straight line that passes through both points. The line represents the graph of the equation \(3x-2y=6\).

Key Concepts

Slope-Intercept Form

One of the crucial concepts in algebra is understanding how to deal with linear equations. The slope-intercept form is particularly handy and is expressed as:
\( y = mx + b \)
where \( m \) represents the slope, and \( b \) represents the y-intercept. This form is beneficial because it directly reveals both the slope and y-intercept, making it simple to graph a line. In the case of the original equation \(3x - 2y = 6\), it was rearranged to give the slope-intercept form of \( y = \frac{3}{2}x - 3 \), clearly indicating the slope as \( \frac{3}{2} \) and the y-intercept as -3. Using this form makes plotting the graph of the linear equation straightforward.

Slope

The slope is a measure of how steep a line is and is indicated by the letter \(m\). Think of it as the rise over the run, or how much the line goes up or down for a certain horizontal distance traveled. For the equation in question, the slope is \(m = \frac{3}{2}\), which means for every two units you move to the right along the x-axis, you will move three units up (for a positive slope) or down (for a negative slope) along the y-axis. This is critical when plotting points because once you know one point on the graph, the slope allows you to find others.

Y-Intercept

The y-intercept of a linear equation is the point where the line crosses the y-axis. This occurs when \(x = 0\). It's marked by the letter \(b\) in the slope-intercept equation. In our example, the y-intercept is -3 (as shown from the slope-intercept form \(y = \frac{3}{2}x - 3\)). This point is super easy to plot since it will always be on the y-axis at \((0, b)\), which in this case is \((0, -3)\). Starting your graph at the y-intercept ensures accuracy in plotting the rest of your points.

Plotting Points on a Graph

Plotting points accurately on a graph is an essential skill. Begin with the y-intercept which you've already identified. Place a dot at \((0, -3)\) on your graph. Next, use the slope to find another point. With a slope of \(\frac{3}{2}\), you'll move 2 units to the right (the run) on the x-axis, and 3 units up (the rise) on the y-axis, landing you at point \((2, 0)\). Plot this point as well. Once you have two points, you can draw a line through them; this line represents the graph of your linear equation. These steps ensure accurate representation of the linear equation on a coordinate plane.