Problem 15 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given points. $$ (-8,6) \text { and }(-13,1) $$

Step-by-Step Solution

Answer

The equation of the line passing through the given points in point-slope form is \(y = -x - 2\)

Step 1: Calculate the Slope

The slope of the line passing through any two points \((-8,6)\) and \((-13,1)\) is given by the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). Substituting the given points into the formula gives \(m = (1 - 6) / (-13 + 8) = -5/5 = -1\). So, the slope of the line is -1.

Step 2: Substitute Point and Slope into Point-Slope Form

The point-slope form of the linear equation is \(y - y_1 = m(x - x_1)\). Substituting the slope \(m = -1\) and one of the points, say \((-8,6)\), we have \(y - 6 = -1 * (x - (-8))\), simplifying gives \(y - 6 = -1 * (x + 8)\).

Step 3: Final Equation

After simplifying, the equation becomes \(y - 6 = -x - 8\). Adding 6 to both sides to isolate y gives the final equation \(y = -x - 2\). The equation represents the line passing through the given points in point-slope form.

Key Concepts

Slope Calculation

When determining the equation of a line in point-slope form, the first step is often to calculate the slope. The slope indicates how steep a line is and is generally represented by the letter \( m \). You calculate the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). In this formula, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line.

As applied in the solution, the given points are \((-8, 6)\) and \((-13, 1)\). Substituting these into the formula gives \[ m = \frac{1 - 6}{-13 + 8} = \frac{-5}{-5} = -1 \].

The slope here is \(-1\), telling us that for every unit we move to the right along the x-axis, the line moves one unit downward along the y-axis.

Linear Equations

Linear equations are mathematical expressions representing straight lines. The primary goal when working with them is usually to derive a formula that describes this line.

You can write a linear equation in various forms, with point-slope form being one of the most direct methods for defining a line. It uses a known point on the line and the slope to specify this line entirely.

The point-slope form is expressed as:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) is a specific point on the line, while \( m \) represents the slope. By substituting the values \((-8, 6)\) for the point and \(-1\) for the slope, we illustrate how the equation becomes \( y - 6 = -1(x + 8) \).

Solving for \( y \) gives the equation in a form that's easier to graph or evaluate: \( y = -x - 2 \). This equation is now ready for use in analyses or graphical representations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate plane. It involves points defined by pairs of numbers that locate them precisely in a plane, making it easier to solve geometrical problems algebraically.

Imagine the coordinate plane as a map, where each point has an address, given by its \( x \)- and \( y \)-coordinates. Problems often involve finding relationships between these points—like a line passing through two known points, as seen in the exercise.

By applying coordinate geometry concepts, you can ascertain how a line traverses this plane using key parameters like slope and intercepts. Once the slope is calculated, and at least one point is identified, you can describe all points on the line using linear equations. This capability not only helps in solving geometry problems but provides foundational tools for fields like physics, engineering, and even in many complex data modeling scenarios.