Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-36: Find the slope-intercept form of the line that satisfies the given conditions. $$ \text { Through } A(5,2) \text { and } B(-1,4) $$

Step-by-Step Solution

Answer

The slope-intercept form is \(y = -\frac{1}{3}x + \frac{11}{3}\).

Step 1: Identify the Slope Formula

To find the slope of the line passing through points \(A(5,2)\) and \(B(-1,4)\), we use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1, y_1) = (5, 2)\) and \((x_2, y_2) = (-1, 4)\).

Step 2: Calculate the Slope

Substitute the coordinates into the slope formula: \[m = \frac{4 - 2}{-1 - 5} = \frac{2}{-6} = -\frac{1}{3}\]. So, the slope \(m\) is \(-\frac{1}{3}\).

Step 3: Use Point-Slope Form

To find the equation of the line, use the point-slope form \(y - y_1 = m(x - x_1)\). Choose point \(A(5,2)\) and substitute \(m = -\frac{1}{3}\), \(x_1 = 5\), and \(y_1 = 2\): \[y - 2 = -\frac{1}{3}(x - 5)\].

Step 4: Simplify to Slope-Intercept Form

Distribute the slope on the right: \[y - 2 = -\frac{1}{3}x + \frac{5}{3}\]. Add 2 to both sides to isolate \(y\): \[y = -\frac{1}{3}x + \frac{5}{3} + 2\].

Step 5: Simplify Further

Convert 2 into a fraction with the same denominator \(\frac{6}{3}\) and add: \[y = -\frac{1}{3}x + \frac{5}{3} + \frac{6}{3}\]. Combine the fractions: \[y = -\frac{1}{3}x + \frac{11}{3}\]. This is the slope-intercept form.

Key Concepts

Slope Calculation

Understanding how to calculate the slope is fundamental in algebra, especially in linear equations. The slope of a line is essentially a measure of its steepness or inclination. In mathematical terms, it's defined as the change in the vertical direction (often referred to as the "rise") divided by the change in the horizontal direction (known as the "run").
To find the slope between two points on a graph, we use the slope formula:

• The formula is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
• Here, \((x_1, y_1)\) are the coordinates of the first point and \((x_2, y_2)\) are the coordinates of the second point.

Let's consider the points \(A(5, 2)\) and \(B(-1, 4)\).
Plug the values into the formula:

• \(y_2 = 4\), \(y_1 = 2\)
• \(x_2 = -1\), \(x_1 = 5\)

The slope \(m\) becomes: \(m = \frac{4 - 2}{-1 - 5} = \frac{2}{-6} = -\frac{1}{3}\).
This shows that the line declines gently as it moves from left to right.

Point-Slope Form

Once we have the slope, we can use the point-slope form of a line to create its equation. This form is especially handy when you know a point on the line and the slope. The point-slope form equation is given by:

• \( y - y_1 = m(x - x_1) \)
• Here, \((x_1, y_1)\) is any known point on the line, and \(m\) is the slope.

For our example, choosing the point \(A(5, 2)\) and the determined slope \(m = -\frac{1}{3}\), substitute into the point-slope form equation:
\[ y - 2 = -\frac{1}{3}(x - 5) \]
This equation helps in moving directly towards the slope-intercept form, providing a straight path to understanding the line's behavior through just one known point and the slope.

Linear Equation

A linear equation in slope-intercept form is a straightforward representation of a line on a graph. It's expressed as \(y = mx + b\), where:

• \(m\) is the slope of the line, indicating its direction and steepness.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

In the given exercise, after applying both the slope and the point-slope form, we simplify to achieve the slope-intercept form.
By distributing \(-\frac{1}{3}\) in \(y - 2 = -\frac{1}{3}(x - 5)\), you get:
\[y - 2 = -\frac{1}{3}x + \frac{5}{3}\]
Adding 2 on both sides, and converting 2 into \(\frac{6}{3}\), gives you:
\[y = -\frac{1}{3}x + \frac{11}{3}\]
This result offers a clear picture of the line, showing how it ascends or descends as the x-values change, and where it interacts with the vertical axis.