Question

Find the slope and y-intercept of each line. Graph the line. $$ 2 x-3 y=6 $$

Short answer

Slope is \(\frac{2}{3}\) and y-intercept is \(-2\).

Step-by-step solution

- Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start by isolating \(y\) on one side of the equation. The given equation is: \[ 2x - 3y = 6 \] First, subtract \(2x\) from both sides: \[ -3y = -2x + 6 \] Next, divide by \(-3\) to solve for \(y\): \[ y = \frac{2}{3}x - 2 \] Now the equation is in slope-intercept form.

- Identify the Slope and Y-Intercept

From the equation \(y = \frac{2}{3}x - 2\), recognize that the slope (\(m\)) is \(\frac{2}{3}\) and the y-intercept (\(b\)) is \(-2\).

- Graph the Line

To graph the line, start by plotting the y-intercept \(-2\) on the y-axis. Next, use the slope \(\frac{2}{3}\) to find another point on the line. From \(-2\) on the y-axis, move up 2 units and to the right 3 units to reach the point \((3, 0)\). Draw a line through these points to complete the graph.

Key concepts

Finding Slope

The slope of a line measures its steepness and direction. In the slope-intercept form of a linear equation, which is written as \(y = mx + b\), the slope is represented by \(m\).
To find the slope from an equation given in standard form, such as \(2x - 3y = 6\), follow these steps:

• First, we need to isolate \(y\) on one side of the equation. Start by subtracting \(2x\) from both sides to get \(-3y = -2x + 6\).

• Next, divide through by \(-3\) to solve for \(y\). This gives us \(y = \frac{2}{3}x - 2\).

This equation is now in the slope-intercept form, where the slope \(m\) is \( \frac{2}{3} \). The slope tells us that for every 3 units we move to the right, we move up 2 units.
Keep in mind that slopes can be positive or negative:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.

Understanding the slope is crucial in graphing and interpreting linear equations.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\).
To find the y-intercept from the equation \(2x - 3y = 6\), we follow the same steps as we did for finding the slope until we have the equation in the form \(y = \frac{2}{3}x - 2\).
Here, we see that the y-intercept \(b\) is \(-2\). This means the line crosses the y-axis at the point \( (0, -2) \).

Steps to identify the y-intercept:

• Rewrite the equation in slope-intercept form, \(y = mx + b\).
• Look at the constant term, \(b\), in this form.

Understanding the y-intercept helps in graphing as it provides a starting point. Plot this point on the y-axis first before using the slope to find other points on the line.

Graphing Linear Equations

Graphing a linear equation involves plotting points on a coordinate plane and drawing a line through them. For the equation \(2x - 3y = 6\), our first step is to convert it to slope-intercept form, which we have done as \(y = \frac{2}{3}x - 2\).

Here are the steps to graph this equation:

• Plot the y-intercept: Start by plotting the y-intercept, which is the point \( (0, -2) \) on the y-axis.


• Use the Slope: From the y-intercept, use the slope to find another point. With a slope of \(\frac{2}{3}\), starting at (0, -2), move up 2 units and right 3 units to get to the point (3, 0).

• Draw the Line: Connect these points with a straight line. This line represents all solutions to the equation \(2x - 3y = 6\).

Remember that the process of graphing helps to visualize the relationship between the variables, making it easier to understand and solve real-world problems involving linear equations.