Problem 52 - 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? Passes through (2,3) and (-6,7)

Step-by-Step Solution

Answer

Based on the given solution, the slope-intercept form, y = -1/2x + 4, might be slightly easier to use as it directly provides the y-intercept value (b) and is a more common form. However, the choice may depend on individual preference and the context in which the equation is being used.

Step 1: Find the slope (m)

To find the slope, use the formula: m = (y2 - y1) / (x2 - x1) Using the given points (2,3) and (-6,7), we have: m = (7 - 3) / (-6 - 2) = 4 / (-8) = -1/2

Step 2: Point-slope form

Using the slope (m = -1/2) and a point (x1, y1) = (2, 3), plug in the values into the point-slope form: y - y1 = m(x - x1) So, y - 3 = -1/2(x - 2)

Step 3: Slope-intercept form

To find the slope-intercept form (y = mx + b), we need to find b using the slope (m = -1/2) and a point (x1, y1) = (2, 3): Substitute the values into the equation and solve for b: 3 = (-1/2)(2) + b 3 = -1 + b b = 4 Now, write the equation in slope-intercept form: y = -1/2x + 4 In this case, both forms are relatively easy to use. However, the slope-intercept form might be slightly easier because it is a more common form and deals directly with the y-intercept value (b). Ultimately, the choice may depend on individual preference and the context in which the equation is being used.

Key Concepts

Point-Slope Form

The point-slope form is a way to express the equation of a line using a specific point and the line's slope. This form is particularly useful when you know one point on the line and the slope, and you need to quickly jot down the equation. The format for the point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) is a known point on the line, and \(m\) is the slope.
This form highlights the slope and a specific point, making it immediate to draw the line on a graph when starting from that point. It can also be a stepping stone for converting to other forms, such as the slope-intercept form.
To use the point-slope form, choose any point on the line; often, the point that is known from a problem or the given data point. Plug in the coordinates of this point and the slope to create your equation. This step is a foundational way to express lines without first calculating the y-intercept.

Slope-Intercept Form

The slope-intercept form is one of the most well-known ways to express a linear equation. It is convenient because it explicitly shows the slope and the y-intercept, which are crucial to understanding the line's behavior. The formula for the slope-intercept form is:

• \( y = mx + b \)

In this equation, \(m\) is the slope, which tells you how steep the line is, and \(b\) is the y-intercept, indicating where the line crosses the y-axis.
This form is straightforward to graph since you can start at the y-intercept and use the slope to find other points. It's also very interpretable; seeing \(b\) directly lets you understand the line's starting point on the y-axis without further calculation.
Converting from other forms, such as the point-slope form, to the slope-intercept form can often make solving graph problems simpler, as it provides immediate visuals of the line's key characteristics. It's particularly easy to use this form when you already have the y-intercept or need to find it.

Finding the Slope

Finding the slope of a line when given two points is a critical skill in understanding linear equations. The slope is essentially a measure of how fast a variable changes, a key concept in mathematics. The formula to find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula calculates the vertical change (rise) over the horizontal change (run) between two points. Simply subtract the y-coordinates from each other, and divide by the subtraction of the x-coordinates.
The concept of slope is essential to understanding how linear equations work and graphing them effectively. It tells you whether a line is rising or falling as it moves from left to right; if \(m\) is positive, the line rises; if negative, the line falls.
Finding the slope is often one of the first steps in creating the equation of a line because it provides insight into the nature and direction of the line based on given data points.