Problem 19 - Algebra Form and Function

Question

Write an equation in point-slope form for the line. Through (6,-7) and (-6,-1)

Step-by-Step Solution

Answer

Answer: The equation of the line is y + 7 = -1/2(x - 6).

Step 1: Find the slope of the line

To find the slope of the line passing through the points (6, -7) and (-6, -1), use the slope formula: m = (y2 - y1) / (x2 - x1). Substitute the coordinates of the given points into the formula: m = (-1 - (-7)) / (-6 - 6)

Step 2: Calculate the slope

Simplify the expression in the numerator and the denominator: m = (6) / (-12) Now, divide the numerator by the denominator to find the slope: m = -1/2 The slope of the line is -1/2.

Step 3: Write the point-slope form equation

Now that we have the slope, we can use either given point to write the equation in point-slope form. Let's use the point (6, -7). The point-slope form is: y - y1 = m(x - x1) Substitute the coordinates of the point (6, -7) and the slope -1/2 into the equation: y - (-7) = -1/2(x - 6)

Step 4: Simplify the equation

Simplify the left side of the equation by eliminating the double negative: y + 7 = -1/2(x - 6) Now, the equation is in point-slope form: y + 7 = -1/2(x - 6) This is the equation of the line that passes through the points (6, -7) and (-6, -1) in point-slope form.

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness and direction. It is denoted by the letter "m" and can be thought of as the "rise over run." In mathematical terms, this means the change in the y-direction (vertical) over the change in the x-direction (horizontal). For any two points on a line, with coordinates \((x_1, y_1)\) and \((x_2, y_2)\), the formula to calculate the slope is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

To calculate the slope of the line through points (6, -7) and (-6, -1), we substitute these into the formula:

• \( m = \frac{-1 - (-7)}{-6 - 6} \)
• Simplifies to: \( m = \frac{6}{-12} \)
• Thus, \( m = -\frac{1}{2} \)

This means the line falls downward from left to right since the slope is negative. Understanding slope is crucial, as it helps in forming the equation of the line, predicting its direction, and indicating how one variable changes with respect to another.

Linear Equations

Linear equations are fundamental in algebra and represent straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables, x, and y, is:

• \( y = mx + b \)

Here, \( m \) represents the slope, and \( b \) is the y-intercept, where the line crosses the y-axis. However, there's another form known as the point-slope form, particularly useful when you have a point on the line and the slope:

• \( y - y_1 = m(x - x_1) \)

In the example provided, we use the point-slope form to write the equation of the line, as we know the slope \(-\frac{1}{2}\) and a point (6, -7) on the line:

• The equation becomes: \( y - (-7) = -\frac{1}{2}(x - 6) \)
• Simplified to: \( y + 7 = -\frac{1}{2}(x - 6) \)

This form highlights how a linear equation can be tailored around any point, offering flexibility in describing lines across various contexts.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to use algebraic methods to solve geometric problems, providing a bridge between algebra and geometry.
Coordinates are used to represent the position of points in space:

• The coordinate plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical).
• A point's position is denoted by an ordered pair \((x, y)\).

In our exercise, two points are given: (6, -7) and (-6, -1). These coordinates are vital for finding the slope and formulating the linear equation. The point-slope form equation uses these coordinates to find and represent the line connecting these points.

• Point (6, -7) becomes our reference for calculating changes in x and y.
• Using the point-slope formula allows us to capture the essence of the relationship between these points on the graph.

Coordinate geometry facilitates a clear understanding of how points, lines, and shapes coexist and intersect in the plane, making it indispensable for solving real-world problems involving space and movement.