Problem 13 - Algebra Form and Function

Question

Write an equation in point-slope form for the line. Through (-1,7) with slope \(m=6\)

Step-by-Step Solution

Answer

Question: Write an equation in point-slope form for a line with a slope of 6 that passes through the point (-1, 7). Answer: The equation in point-slope form is y - 7 = 6(x + 1).

Step 1: Identify the given point and slope

The given point is (-1,7), and the slope, m, is 6. So, we have \((x_1, y_1) = (-1, 7)\) and \(m = 6\).

Step 2: Plug the values into the point-slope form formula

We will now substitute the values of \((x_1, y_1)\) and \(m\) into the point-slope form formula:$$y - y_1 = m(x - x_1)$$Substitute the given values:$$y - 7 = 6(x - (-1))$$

Step 3: Simplify the equation

Simplify the right-hand side of the equation:$$y - 7 = 6(x + 1)$$Now, the equation is in point-slope form with the given point and slope. The final answer is:$$y - 7 = 6(x + 1)$$

Key Concepts

Equation of a Line

An equation of a line is a way to represent a straight line on a coordinate plane using mathematical expressions. It shows how two variables, typically

• x and y, relate to create a position on a two-dimensional surface.

The most common forms of line equations include the slope-intercept form and the point-slope form. The equation involves a slope, which indicates the steepness or angle of the line, and one or more coordinates that anchor the line to the plane.
In this exercise, the focus is on using the point-slope form because it simplifies writing equations when you know a point on the line and the line's slope. This form is expressed as:

• \( y - y_1 = m(x - x_1) \)

Here,

• \( (x_1, y_1) \) represents the coordinates of a known point on the line.
• \( m \) represents the slope.

By plugging in values, you can derive specific equations that represent the line passing through a particular set of coordinates.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, allows us to use algebra to describe geometric concepts. It involves plotting points, lines, and moving objects within a plane defined by a coordinate system.The primary principles at work here include:

• The coordinate plane, which is divided by an x-axis (horizontal) and a y-axis (vertical), forms a grid where each point is defined by a pair of numerical values.
• A coordinate point like \( (-1, 7) \) represents a specific position on this grid where \( -1 \) is the x-coordinate and \( 7 \) is the y-coordinate.

Lines can be drawn by connecting any two points, and their behavior on the coordinate plane is dictated by their mathematical equation. For instance, using the point

• \( (-1, 7) \)

and the slope as guides, one can construct the line demonstrated in the exercise. Understanding the position and movement of lines using coordinate geometry is essential for solving problems that involve spatial relationships.

Slope

The slope of a line is a measure of its steepness and the direction in which it tilts. It describes how much the line rises or falls as it moves horizontally across the plane. You can calculate the slope (

• denoted as \( m \)

) by considering the ratio of the

• vertical change (rise) to the horizontal change (run) between two points on the line. This is often expressed as

• m = \( \frac{\Delta y}{\Delta x} \),

where:

• \( \Delta y \) is the difference in the y-values of two points.
• \( \Delta x \) is the difference in the x-values of those points.

In this exercise, the slope is

• given as \( 6 \).

This means for every unit increase in the x-value, the y-value increases by 6 units. The positive slope indicates that the line rises as it moves from left to right. Understanding slope is crucial for graphing lines and predicting their behavior on the coordinate plane.