Problem 67 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given point and has the given slope. $$ (-1,-4), m=4 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the point (-1,-4) and has a slope of 4 is \( y + 4 = 4(x + 1)\)

Step 1: Identify the given point and slope

The given point is (-1,-4) so, \(x_1 = -1\) and \(y_1 = -4\). The given slope is \(m = 4\).

Step 2: Apply the values into the point-slope form of the line equation

The point-slope form of the line equation is \(y - y_1 = m(x - x_1)\). Substitute the given values to this equation: \(y - (-4) = 4(x - (-1))\).

Step 3: Simplify the equation

Simplify the equation to its simplest form. This results in: \( y + 4 = 4(x + 1)\)

Key Concepts

Linear Equations

Understanding linear equations is foundational for grasping various algebraic and geometric concepts. A linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. These equations typically appear in the form 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.

In our exercise, we were tasked with writing the equation of a line in point-slope form. This specific form, represented as y - y_1 = m(x - x_1), is extremely useful when we know a point on the line (x_1, y_1) and its slope m. It clearly showcases how the slope works in tandem with coordinate points to define the direction and position of the line.

Slope

The slope is a measure of how steep a line is, and it is a critical component of linear equations. Mathematically, slope is defined as the rise over the run -- the change in y over the change in x. The formula for finding the slope between two points is (y_2 - y_1) / (x_2 - x_1). A positive slope means the line is inclined upward, whereas a negative slope indicates a downward incline.

When the slope is a whole number, like in our example m = 4, it can be read as a rise of 4 units up for every 1 unit across to the right. In cases where we have a fraction, the interpretation adapts accordingly. This concept plays a critical role when you're trying to visualize or draw the line on a graph, as well as when you're composing its equation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through the use of coordinates on the Cartesian plane. In this system, each point is defined by a pair of numerical coordinates, which represent its position along the x (horizontal) and y (vertical) axes. Lines, curves, and shapes can be analyzed algebraically using their equations.

For example, the point in the problem, (-1, -4), is located to the left and below the origin of the plane (the point (0,0)), since it has negative values for both coordinates. By understanding how to plot points and interpret these coordinates, we can explore the relationships between algebraic expressions and geometric shapes; such as how a linear equation like the one we are working with models a line.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations that define certain mathematical relationships. They can vary in complexity from simple operations to multifaceted expressions with exponents and multiple variables.

The exercise presented involves the use of an algebraic expression in the form of a linear equation to represent a line. In point-slope form, both the coordinates of a known point and the slope are inserted into an algebraic expression, integrating geometry with algebra. Simplifying algebraic expressions, as done in step 3 of our solution, is often necessary to bring them to a form that is easier to interpret or use further in calculations.