Problem 34 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given point and has the given slope. $$ (-4,3), m=-6 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y = -6x - 21\).

Step 1: Identify the Given Values

Identify the given values in the problem. The given point is (-4,3), so \(x_1 = -4\) and \(y_1 = 3\). The slope, \(m\), is given as -6.

Step 2: Substitute the Values into the Point-Slope Form

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\). Substitute the given values into this equation to get \(y - 3 = -6(x - (-4))\)

Step 3: Simplify the Equation

Simplify the equation. The \(x\) term becomes \(-6 * (x + 4)\) during the simplification process. Further simplification leads to \(y - 3 = -6x - 24\)

Step 4: Rewrite the Equation

Rewrite the above equation in point-slope form: \(y = -6x - 21\)

Key Concepts

Writing Equations of Lines

Understanding how to write equations of lines is a foundational skill in algebra and coordinate geometry. This process involves using the point-slope form of a line, which is expressed as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope of the line, and \(x_1, y_1\) are the coordinates of a specific point the line passes through.

When given a slope and a point, the first step is to insert these values into the point-slope equation. This results in an equation representing the line. The goal is to create a linear equation that accurately describes the relationship between all sets of points along that line. While the point-slope form is already a legitimate form of the line equation, it can be further transformed into other forms, like slope-intercept or standard form, for various purposes.

Slope of a Line

The slope, denoted as \(m\), is a measure of how steep a line is. It's calculated as the rise over run, which means how much the line goes up (or down) for each unit it goes to the right. In the equation \(y - y_1 = m(x - x_1)\), the slope is the multiplier of the \(x\) variable after making substitutions.

The slope informs us about the direction of the line. A positive slope means the line ascends to the right, while a negative slope means it descends to the right. If the slope is zero, the line is horizontal, and if it's undefined (division by zero), the line is vertical. Understanding the slope is essential for writing the equation of a line and predicting the shape of its graph.

Linear Equations

Linear equations form the basis of a vast range of mathematical concepts and real-world problems. In their simplest form, they describe a straight line and can be expressed in multiple ways, including point-slope, slope-intercept (\(y = mx + b\)), and standard form (\(Ax + By = C\)).

The point-slope form is particularly useful when you have a point and a slope, and you want to write the equation of a line quickly. Linear equations in any form, serve as a model to describe a linear relationship between two variables, allowing predictions and deeper analysis of the variables' behavior.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, marries algebra with geometry, allowing the study of geometric shapes using a coordinate system. This field of mathematics is crucial as it enables the analysis of points, lines, and shapes using numeric values and algebraic equations.

Within coordinate geometry, the slope of a line and the point-slope form of an equation are indispensable tools. By knowing just a single point and the slope, one can define the entirety of a line in a two-dimensional plane. This discipline helps demystify complex shapes and is fundamental in many science and engineering fields as well as in everyday applications like mapping and architecture.