- Transform the equation \(-6x - 3y = 12\) to slope-intercept form by isolating \(y\), resulting in \(y = -2x - 4\).
- Identify the slope \(m = -2\) and the y-intercept \(b = -4\).
- Graph the line by plotting the y-intercept at \((0, -4)\) and using the slope \(-2\) to find additional points.
- The slope is \(-2\) and the y-intercept is \(-4\).
Explanation
- Understanding the Problem and Objective
We are given the linear equation \(-6x - 3y = 12\). Our goal is to convert this into the slope-intercept form, which is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. This form allows us to easily read off these two key properties of the line, which are essential for graphing it on a Cartesian coordinate plane.
- Converting to Slope-Intercept Form
To isolate \(y\), we start with the given equation \(-6x - 3y = 12\).
First, add \(6x\) to both sides of the equation to move the x-term to the right:
\[-3y = 6x + 12\]
Next, divide every term on both sides by \(-3\) to solve for \(y\):
\[y = \frac{6x}{-3} + \frac{12}{-3}\]
Simplifying the fractions, we get:
\[y = -2x - 4\]
Now the equation is in the form \(y = mx + b\). Comparing this to our result, we can clearly see that the slope \(m = -2\) and the y-intercept \(b = -4\).
- Graphing the Line
With the slope \(m = -2\) and the y-intercept \((0, -4)\), we can now easily graph the line.
- Start by plotting the y-intercept at the point \((0, -4)\) on the y-axis.
- The slope \(m = -2\), which can be written as \(\frac{-2}{1}\), tells us that for every 1 unit we move to the right on the x-axis, we move 2 units down on the y-axis.
- Starting from the point \((0, -4)\), move 1 unit to the right to reach \(x = 1\), and 2 units down to reach \(y = -6\). This gives us a second point at \((1, -6)\).
- Draw a straight line passing through these two points \((0, -4)\) and \((1, -6)\). The line will also pass through the x-intercept \((-2, 0)\).
Examples
In physics or economics, relationships between variables are often expressed linearly. For example, if you are tracking the distance a car has traveled from a starting point, or the remaining balance in a budget over time, you are essentially dealing with linear equations. By finding the slope and intercept, you can predict future values (the car's position or your remaining budget) at any given time, providing a practical way to manage resources or analyze physical motion.