Problem 112 - Algebra and Trigonometry

Question

Write an equation in point-slope form and slope-intercept form of the line passing through \((-10,3)\) and \((-2,-5)\) (Section \(2.3,\) Example 3 )

Step-by-Step Solution

Answer

The equation of the line passing through points (-10,3) and (-2,-5) is \(y - 3 = -1(x + 10)\) in point-slope form and \(y = -1x - 7\) in slope-intercept form.

Step 1: Calculating the Slope

The formula to calculate the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). Thus, substituting the given points \((-10,3)\) and \((-2,-5)\), the slope can be calculated as follows: \(m = \frac{{-5 - 3}}{{-2 - (-10)}} = \frac{{-8}}{{8}} = -1\).

Step 2: Writing the Point-Slope Form of the Equation

The point-slope form of a linear equation is \(y - y_1 = m(x - x_1)\). Now, select either of the two points, lets select \((-10,3)\), and our calculated slope, -1, we can write the point-slope form as \(y - 3 = -1 (x-(-10))\). Simplifying this equation gives \(y - 3 = -1(x + 10)\).

Step 3: Converting the Equation to Slope-Intercept Form

The slope-intercept form is given by \(y = mx + b\), where b is the y-intercept. To translate our point slope equation \(y - 3 = -1(x + 10)\) to slope intercept form, we simplify the equation to \(y = -1x -7\). Therefore, \(y = -x - 7\) is the equation in slope-intercept form.

Key Concepts

Point-Slope Form

The point-slope form is a straightforward method used to write the equation of a straight line. This form is especially useful when you have a known point on the line and the slope.

Given the formula is:
\( y - y_{1} = m(x - x_{1}) \), where \( (x_{1}, y_{1}) \) is the known point and \( m \) is the slope of the line. To write an equation in point-slope form, insert the slope and the coordinates of the point into the formula.

For instance, given a line passing through the point \( (-10,3) \) with a slope of \( -1 \), the equation would be:
\( y - 3 = -1(x + 10) \).

Point-slope form is versatile and can quickly be manipulated into other forms of linear equations, such as the slope-intercept form, making it a valuable tool in algebra.

Slope-Intercept Form

Slope-intercept form is one of the most commonly used representations of a linear equation and provides a clear view of the slope and y-intercept of a line. The general slope-intercept form is written as:
\( y = mx + b \), where \( m \) represents the slope and \( b \) denotes the y-intercept value.

This form shows how the slope affects the steepness and direction of the line, and the y-intercept shows where the line crosses the y-axis. A positive slope means the line angles upward, while a negative slope indicates it angles downward.

Converting from point-slope to slope-intercept form involves distributing the slope across the x-term and solving for y to isolate it on one side of the equation. For example, from \( y - 3 = -1(x + 10) \), after distribution and simplification, the equation transforms to \( y = -x - 7 \) in slope-intercept form.

Calculating Slope

The slope of a line measures its steepness and is calculated using the coordinates of two distinct points on the line. Using the formula:
\( m = \frac{{y_{2} - y_{1}}}{{x_{2} - x_{1}}} \),

where \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \) are the coordinates of these points, you can compute the slope, \( m \). If the slope is positive, the line rises from left to right; if negative, it falls. A zero slope means the line is horizontal, and an undefined slope indicates a vertical line.

For example, with the points \( (-10,3) \) and \( (-2,-5) \), the slope would be calculated as \( m = \frac{{-5 - 3}}{{-2 - (-10)}} = -1 \), indicating a line that decreases from left to right.

Linear Equations

Linear equations describe straight lines in a coordinate system and come in various forms, each providing different insights about the line's properties. These equations always have one or two variables, typically \( x \) and \( y \), and the highest power is one.

Characteristics of a linear equation include a constant rate of change, which is the slope, and a uniform rise over run ratio between any two points on the line. This constant slope results in the 'straightness' of the line's graph. Linear equations can model real-life situations such as predicting expenses, analyzing trends, and calculating rates. They are foundational in algebra and set the stage for understanding more complex functions and models.