Problem 47 - 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.) \( (-5, 4) \), \( m = 2 \)

Step-by-Step Solution

Answer

The three additional points on the line are \((0, 14), (-2, 10), (2, 18)\).

Step 1: Find the y-intercept

Substitute the known point \((-5, 4)\) and the slope \(2\) into the equation \(y = mx + b\). This gives you a new equation \(4 = 2*(-5) + b\), which simplifies to \(b = 14\).

Step 2: Find Additional Points

Using the equation \( y = 2x + 14 \), select arbitrary values for \(x\) and solve for \(y\). Consider \(x = 0, -2, 2\), the solutions are \( (0,14), \, (-2,10), \, (2,18) \) respectively.

Step 3: Confirm Points

Ensure that all points lie on the line by substituting the x and y values into the line equation \( y = 2x + 14\) and ensuring that both sides hold equal.

Key Concepts

Slope-Intercept Form

When dealing with linear equations in algebra, one of the most common forms you will encounter is the slope-intercept form. This form makes understanding and graphing lines straightforward. The general equation is represented as \( y = mx + b \). Here, \( m \) stands for the slope of the line, indicating how steep the line is. The term \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis.

Understanding the components:

• **Slope (\( m \))**: This tells us the direction and steepness of the line. A positive slope means the line rises as it moves from left to right. A negative slope indicates the line falls.
• **Y-intercept (\( b \))**: Found by setting \( x = 0 \), it's the point where your line will cross the y-axis.

The ease of knowing a line's slope and y-intercept means you can sketch the line quickly, making it invaluable in practical applications like this exercise.

Finding Points on a Line

Once the slope-intercept form of the equation is established, finding additional points on a line becomes an easy task. Let's see how to perform this:

• Start with the slope-intercept form of the equation. In this exercise, the equation is \( y = 2x + 14 \).
• Select different values for \( x \). This choice is arbitrary, but common selections are integers for simplicity.
• Plug each \( x \) value into the equation to solve for \( y \). Each \( (x, y) \) pair is a point on the line.

To illustrate: For \( x = 0 \), inserting into the equation gives \( y = 2(0) + 14 = 14 \), hence the point \( (0, 14) \). By repeating this for multiple \( x \) values, like \( x = -2 \) or \( x = 2 \), you can generate as many points as needed.
Verifying the accuracy of these points involves substituting them back into the equation. If both sides match, the points are confirmed to be on the line.

Coordinate Geometry

In coordinate geometry, we often describe the location and properties of figures using a coordinate system. This system lets us place each mathematical point as an ordered pair \((x, y)\) on a 2D plane.
Coordinate geometry helps visualization by providing a clear visual representation of algebraic equations on a plane.

• Using concepts like the slope-intercept form, we can model lines within this plane and determine intersections, parallelism, and various other relationships.
• Finding points on a line extends beyond mere numbers; it's about understanding how each point relates to the equation and the overall geometric figure.

Connections in coordinate geometry make it easier to transition from abstract algebraic expressions to tangible diagrams, which is why mastering these concepts is fundamental to many scientific fields.