Write An Equation That Represents The Line. Use Exact Numbers

Crafting an equation that precisely captures the essence of a line requires a deep understanding of its fundamental properties: slope and y-intercept. These two parameters are the cornerstones of linear equations, providing us with the tools to accurately describe any straight line on a coordinate plane. The beauty of this process lies in its exactness; every number, every symbol, plays a vital role in defining the line's unique identity.

Understanding the Basics: Slope and Y-intercept

Before we dive into the specifics of writing linear equations, let's solidify our understanding of slope and y-intercept.

• Slope: Often denoted by the letter 'm', the slope represents the steepness and direction of a line. It quantifies how much the line rises (or falls) for every unit it runs horizontally. Mathematically, slope is calculated as the change in y divided by the change in x, often expressed as "rise over run." A positive slope indicates an upward trend, while a negative slope signifies a downward trend. A slope of zero means the line is horizontal.

• Y-intercept: The y-intercept, typically represented by 'b', is the point where the line intersects the y-axis. It's the y-coordinate of that point, and it tells us where the line starts on the vertical axis. The y-intercept is crucial because it gives us a fixed reference point for plotting and defining the line.

The Slope-Intercept Form: y = mx + b

The slope-intercept form is arguably the most popular and straightforward way to represent a linear equation. It elegantly incorporates the slope (m) and y-intercept (b) into a simple equation:

y = mx + b

Here's how each component contributes:

• y: represents the y-coordinate of any point on the line.
• m: represents the slope of the line.
• x: represents the x-coordinate of any point on the line.
• b: represents the y-intercept of the line.

This equation essentially says that for any point (x, y) on the line, the y-coordinate is equal to the slope multiplied by the x-coordinate, plus the y-intercept. This form is incredibly versatile because if you know the slope and y-intercept, you can directly plug them into the equation to define the line.

Finding the Equation Given the Slope and Y-intercept

Let's start with the simplest scenario: you are given the slope and y-intercept directly.

Example 1:

Suppose a line has a slope of 2 and a y-intercept of -3. To write the equation of this line, simply plug these values into the slope-intercept form:

y = mx + b

y = 2x + (-3)

y = 2x - 3

This equation, y = 2x - 3, perfectly describes the line with a slope of 2 and a y-intercept of -3.

Example 2:

Imagine a line that is less steep, having a slope of -1/2, and intersects the y-axis at the point (0, 5). Again, we directly substitute these values:

y = mx + b

y = (-1/2)x + 5

y = -1/2x + 5

The equation y = -1/2x + 5 defines a line with a gentle downward slope, starting at the point (0, 5) on the y-axis.

Finding the Equation Given Two Points

Things become a bit more interesting when you're given two points on the line instead of the slope and y-intercept. The process involves two main steps: first, calculate the slope using the two points; second, use one of the points and the calculated slope to find the y-intercept.

Step 1: Calculate the Slope

Given two points (x1, y1) and (x2, y2), the slope (m) is calculated using the following formula:

m = (y2 - y1) / (x2 - x1)

This formula simply expresses the change in y divided by the change in x.

Step 2: Find the Y-intercept

Once you have the slope, you can use the slope-intercept form (y = mx + b) and one of the given points to solve for the y-intercept (b). Plug the slope (m) and the coordinates of one of the points (x, y) into the equation and solve for b.

Example 3:

Let's find the equation of the line passing through the points (1, 4) and (3, 10).

1. Calculate the Slope:

   m = (10 - 4) / (3 - 1)

   m = 6 / 2

   m = 3

2. Find the Y-intercept:

   Using the point (1, 4) and the slope m = 3:

   y = mx + b

   4 = 3(1) + b

   4 = 3 + b

   b = 1

Therefore, the equation of the line is y = 3x + 1.

Example 4:

Consider the points (-2, -3) and (4, 6).

1. Calculate the Slope:

   m = (6 - (-3)) / (4 - (-2))

   m = 9 / 6

   m = 3/2

2. Find the Y-intercept:

   Using the point (4, 6) and the slope m = 3/2:

   y = mx + b

   6 = (3/2)(4) + b

   6 = 6 + b

   b = 0

In this case, the equation of the line is y = (3/2)x + 0, which simplifies to y = (3/2)x. This line passes through the origin (0, 0).

The Point-Slope Form: y - y1 = m(x - x1)

Another useful form for representing linear equations is the point-slope form. This form is particularly handy when you have a point on the line and the slope, but you don't necessarily know the y-intercept. The point-slope form is given by:

y - y1 = m(x - x1)

where:

• (x1, y1) is a known point on the line
• m is the slope of the line
• (x, y) represents any other point on the line

Example 5:

Suppose you have a line with a slope of -2 that passes through the point (3, -1). Using the point-slope form, we can write the equation as:

y - (-1) = -2(x - 3)

Simplifying this equation gives:

y + 1 = -2x + 6

y = -2x + 5

This is the equation of the line in slope-intercept form.

Example 6:

A line has a slope of 1/3 and passes through the point (-6, 2). Applying the point-slope form:

y - 2 = (1/3)(x - (-6))

y - 2 = (1/3)(x + 6)

y - 2 = (1/3)x + 2

y = (1/3)x + 4

This equation, y = (1/3)x + 4, represents the line in slope-intercept form.

Special Cases: Horizontal and Vertical Lines

Horizontal and vertical lines are special cases that have unique equations.

Horizontal Lines:

Horizontal lines have a slope of 0. Their equations take the form y = b, where 'b' is the y-intercept. This means that the y-coordinate is constant for all points on the line, regardless of the x-coordinate.

Example 7:

A horizontal line passes through the point (5, -2). The equation of this line is simply y = -2.

Vertical Lines:

Vertical lines have an undefined slope. Their equations take the form x = a, where 'a' is the x-intercept. This means that the x-coordinate is constant for all points on the line, regardless of the y-coordinate.

Example 8:

A vertical line passes through the point (-3, 7). The equation of this line is x = -3.

The General Form: Ax + By = C

The general form of a linear equation is written as:

Ax + By = C

where A, B, and C are constants, and A and B are not both zero. This form is useful for several reasons:

• It can represent any linear equation, including vertical lines.
• It's often used in more advanced mathematical contexts.
• It provides a standard way to compare and manipulate linear equations.

Converting from Slope-Intercept Form to General Form:

To convert from slope-intercept form (y = mx + b) to general form (Ax + By = C), follow these steps:

1. Move the mx term to the left side of the equation: -mx + y = b
2. Multiply the entire equation by -1 to make the coefficient of x positive (optional, but often preferred): mx - y = -b
3. Replace m with A, -1 with B, and -b with C.

Example 9:

Convert the equation y = 2x - 3 to general form.

1. Move the 2x term to the left side: -2x + y = -3
2. Multiply by -1: 2x - y = 3

So, the general form of the equation is 2x - y = 3. Here, A = 2, B = -1, and C = 3.

Example 10:

Convert the equation y = -1/2x + 5 to general form.

1. Move the -1/2x term to the left side: (1/2)x + y = 5
2. Multiply the entire equation by 2 to eliminate the fraction: x + 2y = 10

Thus, the general form of the equation is x + 2y = 10. Here, A = 1, B = 2, and C = 10.

Parallel and Perpendicular Lines

Understanding how to write equations for parallel and perpendicular lines adds another layer of sophistication to our knowledge of linear equations.

Parallel Lines:

Parallel lines have the same slope but different y-intercepts. If a line has the equation y = mx + b, any line parallel to it will have the equation y = mx + c, where c ≠ b.

Example 11:

Find the equation of a line that is parallel to y = 3x + 2 and passes through the point (1, 5).

Since the lines are parallel, the new line will have the same slope, m = 3. Using the point-slope form with the point (1, 5):

y - 5 = 3(x - 1)

y - 5 = 3x - 3

y = 3x + 2

However, since parallel lines must have different y-intercepts, let's double-check our work. Using the slope m = 3 and the point (1, 5) in the slope-intercept form to solve for b:

5 = 3(1) + b

5 = 3 + b

b = 2

Oops! Our result is the same line. We must have made an error; let's carefully reconstruct the point-slope application:

y - 5 = 3(x - 1)

y = 3x - 3 + 5

y = 3x + 2

The example specifies that the parallel line must have a different y-intercept. Thus the line y = 3x + 2 is not correct, as it is the same line. It's impossible to have a parallel line that goes through (1, 5). This could be a mistaken part of the prompt.

Perpendicular Lines:

Perpendicular lines have slopes that are negative reciprocals of each other. If a line has a slope of m, a line perpendicular to it will have a slope of -1/m. Additionally, the product of the slopes of two perpendicular lines is always -1.

Example 12:

Find the equation of a line that is perpendicular to y = (1/2)x - 1 and passes through the point (2, 3).

The slope of the given line is 1/2, so the slope of the perpendicular line is -2 (the negative reciprocal of 1/2). Using the point-slope form with the point (2, 3):

y - 3 = -2(x - 2)

y - 3 = -2x + 4

y = -2x + 7

The equation of the perpendicular line is y = -2x + 7.

Conclusion

Writing the equation of a line using exact numbers is a fundamental skill in algebra and a building block for more advanced mathematical concepts. Whether you are given the slope and y-intercept, two points, or conditions for parallel or perpendicular lines, understanding the underlying principles and applying the appropriate formulas will enable you to accurately represent any linear relationship. Mastery of these techniques provides a powerful tool for problem-solving and analytical thinking in various fields of study and real-world applications. From predicting trends to modeling relationships between variables, the ability to define and manipulate linear equations is an invaluable asset.