Problem 18 - Algebra Form and Function

Question

Write an equation in point-slope form for the line. Through (-2,-8) and (2,4)

Step-by-Step Solution

Answer

Question: Write the equation of the line in point-slope form that passes through the points (-2,-8) and (2,4). Answer: y + 8 = 3(x + 2)

Step 1: Find the slope

To find the slope (m) of the line passing through (-2,-8) and (2,4), we can use the slope formula: m = (y2 - y1)/(x2 - x1) Using the given points, we have: m = (4 - (-8))/(2 - (-2))

Step 2: Calculate the slope

Now, we need to simplify the expression to get the actual value of m: m = (4 + 8)/(2 + 2) m = 12/4 m = 3

Step 3: Write the point-slope equation

Now that we have the slope, we can use the point-slope formula to write the equation using one of the two given points (either point can be used). We'll use (-2,-8) for this example: y - y1 = m(x - x1) Plugging in our values, we get: y - (-8) = 3(x - (-2))

Step 4: Simplify the equation

Finally, we can simplify the equation to its final point-slope form: y + 8 = 3(x + 2) So the equation of the line in point-slope form is y + 8 = 3(x + 2).

Key Concepts

Understanding the Slope Formula

The slope formula is a crucial tool in geometry that helps us determine how steep a line is. It is represented as \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( m \) denotes the slope, and \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two distinct points on the line.
To put it simply, the slope measures the rate at which \( y \) changes with respect to \( x \).
In our example, points \((-2, -8)\) and \((2, 4)\) were given, so we used them in the formula like this:

• Subtract the initial \( y \)-value from the final \( y \)-value: \( 4 - (-8) = 12 \)
• Subtract the initial \( x \)-value from the final \( x \)-value: \( 2 - (-2) = 4 \)

This gives the slope as \( m = \frac{12}{4} = 3 \).
The slope of 3 means that for every unit increase in \( x \), \( y \) increases by 3 units.
This positive slope indicates the line rises as it moves from left to right.

Decoding Linear Equations

Linear equations describe straight lines and can take several forms: slope-intercept form, standard form, and point-slope form, among others.
Each form provides a different view of the line’s characteristics.
However, in this particular exercise, we focused on the point-slope form, which is written as \( y - y_1 = m(x - x_1) \).

Point-slope form is especially useful when you know:

• The slope \( m \)
• One point \((x_1, y_1)\)

This form emphasizes the slope and a particular point on the line, making it dynamic to visualize in coordinate geometry.
This particular form is very handy because it can readily be transformed into other common forms of linear equations if needed.
In our step-by-step solution, after determining the slope \( m = 3 \) at Step 3, we substituted the point \((-2, -8)\) into the point-slope equation to find our specific linear equation.

The Role of Coordinates

Coordinates are the pairs of values \((x, y)\) that pinpoint the position of a point in a two-dimensional space.
Every point on a plane can be described using coordinates, establishing a clear numeric location.
In our example, we used the coordinates of the points \((-2, -8)\) and \((2, 4)\) to calculate the slope and formulate our linear equation.

Let's break it down:

• The first value in a coordinate pair, \( x \), tells us how far left or right the point is along the horizontal axis.
• The second value, \( y \), tells us how far up or down the point is along the vertical axis.

These coordinates play a fundamental role not just in creating equations, but also in identifying relationships among different points on a plane. For example, using the slope formula, you can infer how two points are connected, which lets you establish the equation of the line passing through them.
In summary, understanding coordinates allows us to translate geometric concepts into algebraic equations effortlessly.