Question

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((7,-5)\) \(m=-\frac{2}{3}\)

Short answer

The three additional points that the line passes through are \((8, -4)\), \((9, -3)\), and \((10, -2)\).

Step-by-step solution

Initialize the Problem

Observe the provided point, \((7, -5)\), and the provided slope, \(-2/3\). Intention is to use the slope and the known point to find three additional points on the line. This can be achieved by using the point-slope form of the line equation which is \(y - y1 = m(x - x1)\)

Find Point 1

Choose a value for \(x\), for example \(x = 8\). Substitute \(x\) and all the provided values into the line equation - \(-5 - y = -2/3(8 - 7)\). Solving the equation gives \(y = -4\). So, the first additional point is (8, -4)

Find Point 2

Repeat the process used in the previous step but this time let \(x = 9\). Substitute \(x = 9\) into the equation - \(-5 - y = -2/3(9 - 7)\). Solving the equation gives \(y = -3\). Therefore, the second point is \((9, -3)\)

Find Point 3

Repeat the process used in the previous steps but this time let \(x = 10\). Substitute \(x = 10\) into the equation - \(-5 - y = -2/3(10 - 7)\). Solving the equation gives \(y = -2\). Therefore, the third point is \((10, -2)\)

Key concepts

Point-Slope Form

The point-slope form is an essential way of expressing the equation of a line in coordinate geometry. It is especially useful when you have one point on a line and the slope of the line. The formula for the point-slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a known point on the line and \( m \) is the slope of the line.

This form allows us to find the equation of the line and, subsequently, any other point that lies on it. It simplifies the process by requiring just these two elements—a point and the slope—to generate the line's equation. This form is particularly handy in situations like the original exercise, where we're given a point and a slope.

• To use it, substitute the known point and the slope into the formula.
• Solve for \( y \) to determine how the line changes as \( x \) changes.

By following these steps, you can easily extend the line to find additional points that lie on it, much like we did in the exercise.

Linear Equations

Linear equations are foundational to algebra, representing straight lines in a coordinate plane. They are usually written in the form \( ax + by = c \) or more commonly in slope-intercept form \( y = mx + b \). The graph of a linear equation is always a straight line, which makes predicting other points on the line relatively straightforward.

The relationship between \( x \) and \( y \) is constant, governed by the slope \( m \). The slope can tell us how steep the line is and whether it ascends or descends as it moves from left to right. Understanding this relationship is key to manipulating equations and finding unknown points.

• Linear equations can have one or two variables but typically act within the context of two-dimensional spaces.
• Solving a linear equation involves finding all solutions or all points \( (x, y) \) that satisfy the equation.

Therefore, grasping linear equations aids in solving various practical tasks, ranging from simple forecasting in graphs to more complex algebraic problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra with geometry by using a coordinate plane to describe geometric figures. It involves plotting points, lines, and curves to allow for analysis and a deeper understanding of the geometric properties of these figures.

In a coordinate plane, each point is described by an ordered pair \( (x, y) \). This method permits an exact representation of geometric entities and aids in calculating distances, midpoints, and slopes. For the exercise given, understanding how to plot these points and interpret the slope \(-2/3\) is key in finding other points on the line.

• Coordinate geometry allows for precise geometric calculations, including areas and intersections of shapes.
• It bridges the gap between algebraic expressions and geometric visualization.

This branch of mathematics is crucial for describing real-world phenomena where relationships between different quantities need representation on the same plane.