Problem 4 - 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. Slope \(=7,\) passing through \((-4,9)\)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y - 9 = 7(x + 4)\), the slope-intercept form is \(y = 7x + 37\).

Step 1: Calculate the Point-Slope Form

The formula for the point-slope form of a line is given by \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1 ,y_1)\) are the coordinates of the point on the line. Given \(m = 7\) and \((x_1 ,y_1) = (-4, 9)\), the point-slope form of the equation becomes \(y - 9 = 7(x + 4)\).

Step 2: Calculate the Slope-Intercept Form

To transform this into the slope-intercept form (\(y=mx+c\)), distribute the slope through the parenthesis, combine like terms, and then isolate \(y\). The solution goes like this: \(y - 9 = 7x + 28\) simplifies to \(y = 7x + 37\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols; it is a language through which we describe patterns and the relationships between quantities. For instance, when we use algebra to describe the point-slope form of a linear equation, we are translating the geometric properties of a line—its slope and a point it passes through—into an algebraic expression. It allows for the systematic solution of numerical problems, such as finding the equation of a line given certain conditions like a point it passes through and its slope.

Understanding the foundational concept of algebra is crucial for working with linear equations, such as the one described in the example where a given slope and point lead us to dive into the journey of mapping the line's equation. A thorough grasp of algebra not only aids in solving textbook exercises but also enhances logical thinking and problem-solving skills in various real-life applications.

Slope-Intercept Form

The slope-intercept form is an equation of a straight line in the format of \(y = mx + c\). Here \(m\) represents the slope of the line, and \(c\) denotes the y-intercept, the point where the line crosses the y-axis. This form is particularly useful as it provides a clear visualization of the line's steepness and its initial value at the y-axis. In practice, we can obtain the slope-intercept form from the point-slope form by rearranging the equation and solving for \(y\), as demonstrated in the provided exercise.

When we know the slope and a specific point, the slope-intercept form allows us to graph the equation quickly or to calculate the y-value for any given x-value with ease. This form is the go-to tool for many problems in algebra, as it simplifies the process of analyzing linear relationships.

Linear Equations

Linear equations are fundamental in algebra. They represent relations where each variable is to the power of one, creating a straight line when graphed on a coordinate plane. Linear equations come in various forms, including point-slope form and slope-intercept form. The process of moving from the generalized point-slope form, \(y - y_1 = m(x - x_1)\), to the slope-intercept form, \(y = mx + c\), as seen in the exercise, is an invaluable algebraic technique for making equations easier to work with and to graph.

In solving linear equations, we often manipulate them to isolate our variable of interest; in most cases \(y\), as it allows us to explore the relationship between \(x\) and \(y\). Grasping this concept strengthens one's ability to interpret and build models of real-world situations, such as predicting profits, calculating distance, or even determining trends.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using the coordinate plane. It merges algebra and geometry to solve geometric problems. Points are described by ordered pairs of numbers \( (x, y) \), lines are defined by equations, and the coordinate axes come into play for measurement and directions.

The point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\), is an algebraic representation of a line in the coordinate plane. As shown in the example, this form uses the coordinates of a known point on the line, and the slope, to derive the equation governing the line's trajectory through the plane. Understanding coordinate geometry helps in visualizing algebraic expressions and solving geometric problems involving shapes, sizes, and the relative positions of figures.