Problem 20 - Introductory and Intermediate Algebra for College Students 4th

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises \(1-28 .\) Then use the point-slope form of the equation to write the slope-intercept form of the equation. Passing through \((-2,-4)\) and \((1,-1)\)

Step-by-Step Solution

Answer

The slope-intercept form of the line passing through (-2, -4) and (1, -1) is \(y = x - 2\).

Step 1: Find the Slope

Firstly, calculate the slope \(m\) given by the formula \(m = (y_{2} - y_{1})/(x_{2} - x_{1})\). Substituting \((x_{1}, y_{1})\) being \((-2, -4)\) and \((x_{2}, y_{2})\) being \((1, -1)\), the slope \(m\) becomes \((-1 - (-4))/(1 - (-2)) = 3/3 = 1\).

Step 2: Write the Point-Slope Form

Using the point-slope form \(y - y_{1} = m(x - x_{1})\), where \(m = 1\) and \((x_{1}, y_{1})\) is \((-2, -4)\), the equation is \(y - (-4) = 1*(x - (-2))\). Simplifying this yields: \(y + 4 = x + 2\). This can be simplified to \(y = x - 2\).

Step 3: Convert to Slope-Intercept Form

The equation \(y = x - 2\) is already in slope-intercept form \(y = mx + c\). Hence no further steps are required in this case.

Key Concepts

Slope Calculation

To find the equation of a line, we often start with slope calculation. The slope is the measure of the steepness or incline of a line and is denoted by the symbol \( m \). The formula for slope when given two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is:

• \( m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \)

In this exercise, the points given are \((-2, -4)\) and \((1, -1)\). Applying them to the formula gives us:

• \( m = \frac{-1 - (-4)}{1 - (-2)} \)
• This simplifies to \( m = \frac{3}{3} = 1 \)

Hence, the slope of the line passing through these points is 1. Understanding how to calculate slope is crucial as it forms the foundation for finding other forms of the line equation.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a straight line. It is given by the formula:

• \( y = mx + b \)

Where \( m \) is the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
In our solution, after calculating the slope \( m = 1 \), we derived the equation in point-slope form and simplified it to:

• \( y = x - 2 \)

This is already in the slope-intercept form. It's important to note here that:

• The coefficient of \( x \) is the slope \( m \)
• The constant term \( -2 \) is the y-intercept \( b \)

Recognizing this format helps in quickly identifying the slope and the intercept from an equation.

Equation of a Line

An equation of a line can be represented in various forms, each suitable for different calculations. One popular form is the point-slope form, useful when you know one point on the line and the slope. It is expressed as:

• \( y - y_{1} = m(x - x_{1}) \)

Replacing \( m = 1 \), \( x_{1} = -2 \), and \( y_{1} = -4 \) into the equation, we get:

• \( y - (-4) = 1(x - (-2)) \)
• Which simplifies to \( y + 4 = x + 2 \)

From this, simplifying further leads to \( y = x - 2 \), translating it smoothly into the slope-intercept form. Understanding these conversions between forms empowers you to handle various problem types in algebra and geometry confidently.