Problem 48 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Slope \(-2, y\) -intercept (0,-4)

Step-by-Step Solution

Answer

The equation is \( y = -2x - 4 \).

Step 1: Identify the Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept.

Step 2: Insert Given Values

Replace \( m \) with \(-2\) and \( b \) with \(-4\) into the slope-intercept form equation. This gives us the equation \( y = -2x - 4 \).

Step 3: Verify the Equation

Check the equation by understanding that the slope \( m = -2 \) and the y-intercept \( b = -4 \) matches the given conditions of the problem, confirming that the equation \( y = -2x - 4 \) is correct.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most convenient ways to write an equation of a line. It is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, or where the line crosses the y-axis.

This form is particularly helpful for quickly identifying these two key features of a line:

• It allows us to easily see the rate of change, or steepness, of the line (the slope).
• It clearly shows where the line crosses the vertical y-axis (the y-intercept).

By understanding the components of the slope-intercept form, you can effortlessly write the equation of a line, if you know the slope and y-intercept.

Slope

The slope is a measure of how steep a line is. In the slope-intercept form \( y = mx + b \), the slope is represented by \( m \). It shows the amount of change in the y-value for a one-unit change in the x-value.

In mathematical terms, the slope \( m \) is calculated as the rise over run, or the change in y divided by the change in x:

• A positive slope means the line is increasing (going uphill) as you move from left to right.
• A negative slope means the line is decreasing (going downhill) as you move from left to right.
• A slope of zero indicates a horizontal line, showing no change in y with changes in x.

In our exercise, the slope was given as \(-2\), suggesting the line decreases by 2 units in y for each increase of 1 unit in x.

Y-Intercept

The y-intercept is where the line crosses the y-axis. This occurs when the x-value is zero. In the slope-intercept equation \( y = mx + b \), \( b \) is the y-intercept.

The y-intercept provides a starting point for the line on the graph:

• If \( b \) is positive, the line crosses the y-axis above the origin.
• If \( b \) is negative, it crosses below the origin.

In our example, the y-intercept given is \(-4\), indicating that the line crosses the y-axis at the point (0, -4). This means the line starts 4 units below the origin on the vertical axis.

Math Problem-Solving

Solving math problems involves understanding the key components of the problem and systematically applying concepts to find the solution. When dealing with linear equations and the slope-intercept form, follow these steps:

1. **Identify What You Know**: Start by noting the given slope and y-intercept.
2. **Apply the Right Formula**: Use the slope-intercept form \( y = mx + b \) to plug in the slope and y-intercept.
3. **Verify**: Once you've written the equation, check your work by comparing it to the problem's conditions.

This process helps ensure that your solution not only uses the correct formula but also aligns with the problem's requirements. For instance, in our problem, recognizing the slope as \(-2\) and the y-intercept as \(-4\) directly leads to the equation \( y = -2x - 4 \), which can be verified by the given points.