Problem 15 - Precalculus

Question

Find an equation for the line having the given slope and passing through the given point. Write your answers in the form \(y=m x+b\). (a) \(m=-5 ;\) through (-2,1) (b) \(m=\frac{1}{3} ;\) through \(\left(-6,-\frac{2}{3}\right)\)

Step-by-Step Solution

Answer

(a) \( y = -5x - 9 \); (b) \( y = \frac{1}{3}x + \frac{4}{3} \)

Step 1: Understand the Point-Slope Form Equation

The point-slope form of the equation of a line is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line. We will use this to find the specific line equation.

Step 2: Plug values into Point-Slope Form for Part (a)

For part (a), we have \( m = -5 \) and the point is \((-2, 1)\). Substitute these values into the point-slope form equation: \( y - 1 = -5(x + 2) \).

Step 3: Simplify to Slope-Intercept Form for Part (a)

Distribute \(-5\) in the equation: \[ y - 1 = -5x - 10 \].Add \(1\) to both sides to solve for \( y \): \[ y = -5x - 9 \]. This is the equation in the slope-intercept form \( y = mx + b \).

Step 4: Plug values into Point-Slope Form for Part (b)

For part (b), \( m = \frac{1}{3} \) and the point is \((-6, -\frac{2}{3})\). Substitute these into the point-slope form: \[ y - \left(-\frac{2}{3}\right) = \frac{1}{3}(x + 6) \].

Step 5: Simplify to Slope-Intercept Form for Part (b)

First, simplify the equation:\[ y + \frac{2}{3} = \frac{1}{3}x + 2 \].Subtract \(\frac{2}{3}\) from both sides to solve for \( y \):\[ y = \frac{1}{3}x + \left(2 - \frac{2}{3}\right) \].Simplify further: \[ y = \frac{1}{3}x + \frac{4}{3} \].

Key Concepts

Point-Slope Form

The point-slope form is a fundamental way to express a linear equation. It is particularly useful for constructing an equation when you already know the slope of the line and a specific point that it passes through. The general formula is given by: \( y - y_1 = m(x - x_1) \).
This format allows you to visually see how a line swings around a particular point.

• \( m \) is the slope of the line. It's the value that tells you how steep the line is.
• \((x_1, y_1)\) represents the coordinates of a known point on the line.

Using the point-slope form is like finding a way to draw a line across a graph. It shows you both direction (through the slope) and location (through the point).
Plugging specific values into this formula can help you transition into writing your equation in more commonly seen formats, such as the slope-intercept form.

Slope-Intercept Form

The slope-intercept form is perhaps the most well-known way to represent linear equations. It looks like this: \( y = mx + b \).
This form nicely outlines two key pieces of information about the line:

• \( m \) is still the slope, just like in the point-slope form, so it tells us how fast or slow the line ascends or descends.
• \( b \) is the y-intercept, which shows where the line crosses the y-axis.

Once you have transformed your equation into the slope-intercept form, it’s easy to grasp both the steepness of the line (whether it moves upwards or downwards) and exactly where it starts off on the y-axis when \( x = 0 \).
If you ever have a linear equation or you need to graph it, you'll want it in this form. It's straightforward and immediately tells you how the line behaves.

Slope of a Line

The slope of a line is essential for understanding how one variable changes in relation to another. Slope is often defined as "rise over run," which is a way to quantify the vertical change per unit of horizontal change. Mathematically, it's represented as \( m \).
For a line specified by two points \((x_1, y_1)\) and \((x_2, y_2)\), slope \( m \) is calculated by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]There are a few things to note:

• A positive slope means the line ascends from left to right.
• A negative slope indicates descent from left to right.
• A zero slope represents a perfectly horizontal line, showing no vertical change.
• An undefined slope (where the denominator of the slope formula is zero) signifies a vertical line.

Understanding slope is like reading a line's steepness in a graphic story. Every line’s slope gives away its "character" of how sharply it angles across a coordinate plane.