Problem 51 - Algebra Form and Function

Question

Is the point-slope form or slopeintercept form the easier form to use when writing an equation for the line? Slope \(=3\), Intercept \(=-6\)

Step-by-Step Solution

Answer

Answer: The slope-intercept form is easier to use when writing an equation for a line given its slope and y-intercept.

Step 1: Point-Slope Form

The point-slope form of a linear equation is: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is any point on the line and \(m\) is the slope of the line. In our case, the slope is given as \(m = 3\). We don't have a specific point, but since we have the y-intercept as \(-6\), we know that the line passes through the point \((0, -6)\). Therefore, we can write the equation in the point-slope form as: \(y - (-6) = 3(x - 0)\), which simplifies to: \(y + 6 = 3x\).

Step 2: Slope-Intercept Form

The slope-intercept form of a linear equation is: \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. In our case, we already have the slope given as \(m = 3\) and the y-intercept as \(b = -6\). So the equation in slope-intercept form is: \(y = 3x - 6\).

Step 3: Comparing the Ease of Use

In this specific case, we can easily write the equation of the line directly in the slope-intercept form, as both the slope and y-intercept values are given. On the other hand, while we can also use the point-slope form, it requires us to identify an appropriate point on the line and then substitute the values into the equation, resulting in a slightly more cumbersome process. Therefore, the slope-intercept form is the easier form to use when writing the equation of the line in this case.

Key Concepts

Point-Slope Form

The point-slope form is a highly versatile format for expressing linear equations. It’s written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is any known point on the line, and \(m\) is the slope. This form is particularly useful when you have a specific point and the slope of the line.
Here's a breakdown of key components:

• \((x_1, y_1)\): A known point on the line. Even if you know only one point, you can substitute it here.
• \(m\): The slope of the line, tells you how steep the line is.

In our solution, since the y-intercept \(-6\) is given, we used the point \((0, -6)\). Plugging in these values, the equation becomes \(y + 6 = 3x\). Simplifying it can also help you transition to other forms, like slope-intercept, if needed.

Slope-Intercept Form

Slope-intercept form \(y = mx + b\) is straightforward and often easier to use than point-slope, especially when both the slope and y-intercept are already known. Here, \(m\) is the slope, and \(b\) is where the line crosses the y-axis.
Here's why it's popular:

• Immediate graphing: Allows you to quickly determine how the line behaves on a graph.
• Direct form: If you have \(m\) and \(b\), it's as simple as plugging these values in to get the full equation.

In this problem, with \(m = 3\) and \(b = -6\), you straightforwardly write \(y = 3x - 6\). This streamlined formula immediately reveals how the line cuts through the y-axis at \(-6\) with a slope of \(3\), changing \(y\) by \(3\) units for each unit increase in \(x\).

Y-Intercept

The y-intercept is quite literally the point where your line crosses the y-axis. When expressed in the equation of a line, it's represented as the constant \(b\) in the slope-intercept form \(y = mx + b\).
Here's what makes the y-intercept important:

• Starting point: It serves as a starting reference on the graph.
• Quick identification: Easily identifies where the line will intersect the y-axis, helping in sketching the graph quickly.

In this exercise, the y-intercept is given as \(-6\), meaning the line crosses the y-axis at the point \((0, -6)\). It shows the height of the line when \(x\) is zero, making it crucial for plotting lines and understanding their behavior relative to the origin.