Problem 20 - 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. $$ (-1,-3), m=4 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y = 4x + 1\).

Step 1: Identify the Given Data

We are given the point \((-1,-3)\) and the slope \(m = 4\). We can represent the point as \((x_1, y_1) = (-1,-3)\) and the slope as \(m = 4\). These are the values we will put into the formula.

Step 2: Write down the Point-Slope Form

The point-slope form of a line is represented by the equation \(y - y_1 = m(x - x_1)\). The variables \((x_1, y_1)\) are the coordinates of the given point and \(m\) is the slope of the line.

Step 3: Substitute the Values and Simplify

We substitute the values we have into the formula, which gives us \(y - (-3) = 4(x - (-1))\). Simplifying this, we get \(y + 3 = 4(x + 1)\). The equation hence becomes \(y + 3 = 4x + 4\).

Step 4: Finalize the Equation

To finalize the equation, we isolate \(y\) by subtracting 3 from both sides, which gives us \(y = 4x + 4 - 3\). Thus, the equation of the line in point-slope form is \(y = 4x + 1\).

Key Concepts

Linear Equations

When dealing with lines in coordinate geometry, linear equations are essential. A linear equation describes a straight line, and its general format is \( Ax + By + C = 0 \). Here, \( A \), \( B \), and \( C \) are constants. In simple terms, a linear equation is an algebraic equation in which each term is either a constant or a product of a constant and a single variable.

They model relationships where the quantities involved change at a constant rate. Specifically, any equation that can be translated into this form is linear:

• They have variables raised to the power of one, no higher.
• No variable in a linear equation is multiplied by another variable.
• No variables can be within a square root, exponent, or denominator in standard linear equations.

The point of a linear equation is that the graphical representation of the solutions is a straight line, which is what makes them fundamentally different from quadratic or cubic equations, for example.

Slope-Intercept Form

The slope-intercept form is one of the most user-friendly ways to express a linear equation. It's written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, or where the line crosses the y-axis. This form is quite useful for quickly sketching a graph or identifying important features of the line.

The components are straightforward:

• The slope, \( m \), indicates the steepness and direction of the line. A positive slope moves upward from left to right, while a negative slope moves downward.
• The y-intercept, \( b \), is where the line crosses the y-axis. This tells you the value of \( y \) when \( x \) is 0.

For example, in the equation \( y = 4x + 1 \) obtained from the original exercise, the slope \( m \) is 4, showing how steeply the line ascends, while the y-intercept \( b \) is 1, showing the point at which the line crosses the y-axis.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra with geometry, allowing you to study geometric figures using a coordinate plane. Points in this plane are identified by their x and y coordinates. For example, the point \((-1,-3)\) represents a position on this 2D plane, where \(-1\) is the x-coordinate, and \(-3\) is the y-coordinate.

In coordinate geometry, linear equations help describe lines. These lines are defined by points and slopes:

• Using points like \((x_1, y_1)\), you can determine the position of the line.
• Using the slope \( m \), you determine the line's orientation.

One practical application in this type of geometry is determining the distance between two points or finding the midpoint of a segment defined by two endpoints. It provides a robust method for proving geometric theorems and for solving real-world problems involving distances and other measurements.