Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) Point (-1,-6) Slope \(m=-\frac{1}{3}\)

Step-by-Step Solution

Answer

The additional points on the line are (0,-\(\frac{17}{3}\)), (1,-\(\frac{20}{3}\)), (2,-\(\frac{23}{3}\)).

Step 1: Find the y-intercept

We insert the given point \((-1,-6)\) and the slope \(m = -\frac{1}{3}\) into the equation for a straight line, \(y = mx + c\). This gives us: \(-6 = -\frac{1}{3}(-1) + c\). Solving for \(c\) we get \(c=-6+\frac{1}{3}=-\frac{17}{3}\).

Step 2: Select Arbitrary x-values

Choose three x-values. These can be any value, for instance we can select 0, 1 and 2.

Step 3: Compute Corresponding y-values

Compute the corresponding y-values using the equation \(y = mx + c\). Substituting \(c=-\frac{17}{3}\), \(m=-\frac{1}{3}\) and x-values (0,1,2), we get three new points: (0,-\(\frac{17}{3}\)), (1,-\(\frac{20}{3}\)), (2,-\(\frac{23}{3}\)).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a simple way to represent a straight line on a graph. It is expressed as \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept. The slope \( m \) describes the steepness or incline of the line, while the y-intercept \( c \) indicates where the line crosses the y-axis. To find the equation of a line using this form, you need two pieces of information: a specific point on the line and the slope. Using the given point

• (-1, -6)

and the slope

• \( m = -\frac{1}{3} \)

you can substitute them into the equation to solve for \( c \), finding the y-intercept as shown in the original example. This allows you to define the line's equation, which is crucial for determining additional points on the line.

Linear Equations

Linear equations are mathematical expressions that create straight lines when plotted on a graph. These equations only involve constants and a variable raised to the first power. When you solve linear equations, you are essentially finding the points that make the equation true.The general form of a linear equation is \( Ax + By = C \), but it's often more useful to write it in the slope-intercept form \( y = mx + c \). This format immediately shows us the slope and y-intercept, making graphing and understanding the line easier. In the context of the original exercise, by inserting the given slope and point into the slope-intercept form, you arrive at the equation that represents the line. This equation helps identify other points on the line, which are all solutions to this linear equation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebraic methods to solve geometric problems. It merges algebra and geometry through the use of coordinate planes, which allows you to solve problems involving shapes and lines.In a coordinate plane, each point is described by an \( x \)-coordinate and a \( y \)-coordinate. The original exercise asks you to find points on the line, which involves understanding how the slope affects the position of each point. By using coordinate geometry principles, you can select various \( x \)-values to calculate corresponding \( y \)-values, as demonstrated in the step-by-step solution. This approach not only finds specific points but also helps visualize how the line behaves within the coordinate plane, making it a powerful tool for understanding linear relationships.