Problem 49 Use the y-intercept and slope to... [FREE SOLUTION] (2024)

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The slope-intercept form of a linear equation is one of the most common and easiest ways to represent a line. This form is written as \(y = mx + b\). Here’s a breakdown of each component:

• \(y\): This is the dependent variable or the value on the y-axis.
• \(m\): This represents the slope of the line. The slope indicates how steep the line is, and it is defined as the 'rise' over 'run' or \(\frac{\Delta y}{\Delta x}\).
• \(x\): This represents the independent variable or the value on the x-axis.
• \(b\): This is the y-intercept. It’s where the line crosses the y-axis when \(x = 0\).

In the given equation \(y = -3x + 1\), the slope \(m\) is -3 and the y-intercept \(b\) is 1. By recognizing this form, we can plot the line easily.

The y-intercept of a line is the point where the line crosses the y-axis. It’s crucial for plotting a graph because it gives us a starting point.
In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\).

For our equation \(y = -3x + 1\), the y-intercept is 1. This means the line crosses the y-axis at the point (0,1).

To plot the y-intercept:

• Find the point where the line will intersect the y-axis, which is where \(x = 0\).
• In our equation, when \(x = 0\), \(y = 1\).
• So, we plot the point (0,1) on the graph.

This gives us the initial point to draw our line, and in the next step, we will use the slope to find other points.

Graphing a line involves plotting points and then drawing a line through those points. To graph lines using the slope and y-intercept method, follow these steps:

• Identify the slope \(m\) and the y-intercept \(b\).
• Plot the y-intercept point on the graph.
• Use the slope to find another point on the line. Recall that the slope \(m\) can be seen as \(\frac{rise}{run}\).
• Starting from the y-intercept, use the slope. For \(y = -3x + 1\), the slope is -3, meaning for every 1 unit you move right (run), move 3 units down (rise).
• Plot the second point using slope calculations, in this case, starting at (0,1) and moving to (1,-2).

Finally, draw a straight line through these points and extend it in both directions to complete your graph.
Graphing lines with the slope-intercept method is straightforward and a fundamental skill in algebra.