Problem 101 - Algebra and Trigonometry

Question

Write the point-slope form and the slope-intercept form of the line passing through \((-1,-2)\) and \((-3,4) .\) (Section 2.3 Example 3 )

Step-by-Step Solution

Answer

The point-slope form of the line is \(y + 2 = -3(x + 1)\) and the slope-intercept form of the line is \(y = -3x - 5\).

Step 1: Calculate the slope

The slope (m) of a line is calculated using two points \((x_1, y_1)\) and \((x_2, y_2)\), using the formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). In this case, \(x_1 = -1, y_1 = -2, x_2 = -3 , y_2 = 4\). So, \(m = \frac{{4 - (-2)}}{{-3 - (-1)}} = \frac{6}{-2} = -3 .\)

Step 2: Use point-slope form

The point-slope form of a line is \(y - y_1 = m(x - x_1)\). Substitute \(m = -3\), \(x_1 = -1\), and \(y_1 = -2\) into this formula to get \(y - (-2) = -3(x - (-1))\), which simplifies to \(y + 2 = -3(x + 1)\).

Step 3: Convert to slope-intercept form

Rearrange the point-slope form from Step 2 into the slope-intercept form \(y = mx + b\). Here, distribute the slope -3 to \(x + 1\) and subtract 2 from both sides. This gives \(y = -3x - 3 - 2 = -3x - 5\).

Step 4: Check Results

Check that the lines \(y + 2 = -3(x + 1)\) and \(y = -3x - 5\) pass through the points \((-1,-2)\) and \((-3,4) .\)

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express linear equations. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis. Understanding this form is crucial as it allows for quick graphing of linear equations and immediate identification of the line's characteristics.

For example, in our problem, after determining the slope of the line to be -3, we rearranged the point-slope equation to get \( y = -3x - 5 \). This means our line has a slope of -3 and crosses the y-axis at the point (0, -5). Being adept with converting between point-slope and slope-intercept forms can significantly help ease the process of analyzing and graphing lines.

Calculate the Slope of a Line

Calculating the slope of a line that connects two points \((x_1, y_1)\) and \((x_2, y_2)\) is fundamental to understanding linear relationships. The slope, denoted as \( m \), is an expression of how steep a line is and it is calculated by the formula \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \).

In simpler terms, the slope tells us how much the y-value of a line changes with a unit change in the x-value. In our problem, the slope is calculated as \( m = -3 \) which indicates a steep, negative incline. A positive slope would mean an upward incline, while a zero slope signifies a horizontal line. Thus, mastering slope calculation is crucial for understanding the direction and steepness of linear equations.

Linear Equations

Linear equations form the basis for a wealth of mathematical concepts and real-world applications. In their simplest form, these equations represent straight lines when plotted on a coordinate plane. A basic linear equation involves two variables, x and y, and can be expressed in various forms, including standard form \( Ax + By = C \), point-slope form, and slope-intercept form.

In our problem, we've worked with the point-slope and slope-intercept forms - transforming from one to the other using algebraic manipulation. Understanding how these forms relate and how to move between them gives students the tools to tackle a wide range of problems related to lines, whether it's in geometry, algebra, or beyond.