Problem 9 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=-1,\) passing through \(\left(-\frac{1}{2},-2\right)\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -x -3/2\).

Step 1: Writing point-slope form

We substitute \(m= -1\), \(x1 = -1/2\), and \(y1 = -2\) into the point-slope formula, which gives \(y - (-2) = -1(x - (-1/2))\), simplifying to \(y + 2 = -1(x + 1/2)\). By distributing -1 to the terms in the parentheses we get \(y + 2 = -x -1/2\).

Step 2: Converting to slope-intercept form

The slope-intercept form equation is constructed by isolating \(y\). We achieve this by subtracting 2 from both sides of our equation from step 1. Thus our result is \(y = -x -1/2 - 2\). Further simplifying by combining like terms gives the final equation: \(y = -x -3/2\).

Key Concepts

Algebra

Algebra is a branch of mathematics that utilizes symbols and letters to represent numbers and quantities in formulas and equations. The key to understanding algebra is knowing that these symbols, often called variables, can stand for unknown values that we can solve for. For instance, in our exercise, variables like `m` represent the slope of a line, while `x1` and `y1` represent specific points that the line passes through.

The ability to manipulate these variables using algebraic principles allows us to formulate relationships between quantities and find solutions to problems that may be too complex to solve by simple arithmetic. For students approaching linear equations, mastery of algebraic manipulation is essential in transitioning from point-slope form to slope-intercept form.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to represent a line in coordinate geometry. It has the formula `y = mx + b`, where `m` is 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 because it immediately gives us valuable information about the line without having to perform additional calculations. The slope tells us how steep the line is, and the y-intercept indicates the starting point of the line on the graph. Converting from point-slope form, as shown in our exercise, involves algebraic manipulation to achieve the slope-intercept format, which can make graphing and analyzing the line much simpler.

Linear Equations

Linear equations form the foundation for much of algebra and coordinate geometry. These equations describe lines in a two-dimensional space, characterized by their constant slope and straight paths. The general forms for these equations include the point-slope form, which is `y - y1 = m(x - x1)`, and the slope-intercept form which we addressed in the previous section.

In the given exercise, the linear equation takes the form of a line with a slope of `-1` - this means the line descends one unit vertically for every unit it moves horizontally. As a student works through algebraic steps to manipulate these forms of equations, they deepen their understanding of the relationships between the algebraic representations and their corresponding graphical depictions.

Coordinate Geometry

Coordinate geometry, sometimes known as analytic geometry, merges the numerical rigor of algebra with the visual impact of geometric shapes. It provides a system for describing geometrical shapes in a numerical way that can be analyzed algebraically. The coordinate plane, defined by perpendicular x and y axes, allows us to plot points, lines, and curves using ordered pairs like (x, y).

Every line in the coordinate plane can be expressed through an equation, and understanding these equations in their different forms, such as point-slope or slope-intercept, is vital. Our exercise works directly with coordinate geometry, as we take a point on the line and the slope of the line to derive its equation, showing the bridge between numerical information and graphical representation.