Problem 43 - Precalculus with Limits

Question

In Exercises 41-50, 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.) \( (5, -6) \), \( m = 1 \)

Step-by-Step Solution

Answer

The three additional points on the line with point (5,-6) and slope 1 are (0,-11), (2, -3), and (7,1).

Step 1: Understand the slope-point form of a line

The slope-point form of a line is given by \(y - y_1 = m(x - x_1)\), where m is the slope of the line and \((x_1, y_1)\) is the known point on the line. Here, the known point is (5, -6) and slope m is 1.

Step 2: Substitute x-values and solve for y

Choose different x-values and substitute them into the equation to find corresponding y-values. For instance:- When \(x = 0\), \(y - -6 = 1 * (0 - 5) => y = -6 - 5 = -11\). So, one additional point is \((0, -11)\).- When \(x = 2\), \(y - -6 = 1 * (2 - 5) => y = -3\). So, another point is \((2, -3)\).- When \(x = 7\), \(y - -6 = 1 * (7 - 5) => y = 1\). Thus, a third point is \((7,1)\).

Step 3: Summarize the additional points

The three additional points through which the line passes are (0, -11), (2,-3), and (7, 1).

Key Concepts

Point-Slope Form

The point-slope form is a powerful tool in coordinate geometry. It provides a way to describe a line when you know a point on the line and the slope. This form of the equation is given by \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope of the line, and \((x_1, y_1)\) is the given point through which the line passes.

• The **slope** \( m \) is a measure of the line's steepness. It shows the ratio of vertical change (rise) to horizontal change (run) between two points.
• In our example, the slope is 1, indicating a 45-degree angle of incline, as for each step right, the line goes one up.

Once you have the point and the slope, substituting these values into the point-slope form allows you to find the y-values for any x-value. This flexibility makes it easy to sketch the line or calculate additional points quickly.

Linear Equations

A linear equation is any equation that forms a straight line when it is graphed on a coordinate plane. The essence of linear equations is that they depict a constant rate of change. This uniform rate is revealed in the equation’s slope, which remains constant as you move along the line.

• Each linear equation has a general form, such as:\[ y = mx + c \], where \( c \) is the y-intercept.
• The equations can also be written in point-slope form, which is directly related to the method used in this exercise to find new points.

While sometimes you may need to convert between forms to understand the equation better or apply it to solve real-world problems, the core idea is that linear equations help model predictable relationships between variables. By choosing x-values and using these forms, you can easily find corresponding y-values and plot the linear behavior.

Coordinate Geometry

Coordinate geometry integrates algebra and geometry to describe geometric figures using a coordinate system. By employing coordinates, it allows for precise representations and calculations based on algebraic formulas.

• In this system, every point on the plane is described as an ordered pair \((x, y)\), which can be used to locate it precisely.
• Using a known point and slope, coordinate geometry makes it easier to understand linear relationships and solve geometrical problems.

Here, we see it in action where the point-slope form allows calculations of new points (other locations) on the same line, using one known point \((5, -6)\). This method is central to many applications in geometry and real-world scenarios, illustrating how separate disciplines come together to solve problems efficiently. Utilizing coordinate geometry in such exercises not only helps visualize but also mathematically verify the characteristics of a line.