Question

Find the slope and y-intercept of each line. Graph the line. $$ 3 x+2 y=0 $$

Short answer

The slope is -3/2 and the y-intercept is 0.

Step-by-step solution

- Rearrange the equation into slope-intercept form

Start by rearranging the equation into the form of \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. The given equation is \(3x + 2y = 0\). Subtract \(3x\) from both sides: \(2y = -3x\). Now, divide everything by \(2\) to isolate \(y\): \(y = -\frac{3}{2}x\).

- Identify the slope and y-intercept

The slope-intercept form of the equation obtained is \(y = -\frac{3}{2} x + 0 \). Identify the slope (\(m\)) and y-intercept (\(c\)). Here, \(m = -\frac{3}{2}\) and \(c = 0\).

- Graph the line

To graph the line, start at the y-intercept \((0, 0)\). Use the slope \(-\frac{3}{2}\) to determine another point. From \((0, 0)\), move 3 units down and 2 units to the right (since the slope is negative), which lands you at \((2, -3)\). Draw the line through \((0, 0)\) and \((2, -3)\).

Key concepts

slope-intercept form

The slope-intercept form is one of the most popular ways to express a linear equation. The general format is given by: \(y = mx + c\) where:
- \(m\) represents the slope of the line
- \(c\) represents the y-intercept.
The y-intercept is the point where the line crosses the y-axis. This form makes it easy to quickly identify both the slope and y-intercept of a line, which is very helpful when graphing linear equations.

graphing linear equations

Graphing linear equations involves plotting the line on a coordinate plane. Begin by identifying the y-intercept and plot this initial point on the y-axis.
Next, use the slope to find another point. For example, if the slope is \(-\frac{3}{2}\), start from the y-intercept and move down 3 units and to the right 2 units—this gives you a second point.
Draw a line through these two points extending it in both directions, showing the full extent of the line.
This visual representation helps you understand the relationship between the variables and the behavior of the equation.

slope calculation

The slope of a line is a measure of its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between any two points on the line. Mathematically, the slope \(m\) is given by:

• m = \frac{(change\ in\ y)}{(change\ in\ x)}\

In this exercise, the equation is rewritten into the Y form, and the coefficient of x, \(-\frac{3}{2}\), gives us the slope. This tells us that for every 2 units we move right along the x-axis, we move 3 units down along the y-axis.

y-intercept determination

The y-intercept of a line is the point where it crosses the y-axis. In the slope-intercept form \(y = mx + c\), \(c\) represents the y-intercept. In the rearranged equation from the exercise \(y = -\frac{3}{2}x + 0\), the \(c\) value is \0\. This tells us that the line crosses the y-axis at the origin (0, 0). When graphing a line, starting at this point often simplifies the plotting process.