Problem 10 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given points. $$ (0,0) \text { and }(-6,-5) $$

Step-by-Step Solution

Answer

The equation of the line given the points (0,0) and (-6,-5) in point-slope form is y = (5/6)*x

Step 1: Find the Slope of the line

Let's first define the coordinates of the two given points, we have Point1 as (x1, y1) = (0, 0) and Point2 as (x2, y2) = (-6, -5). We can calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Substituting the values results in m = (-5 - 0) / (-6 - 0) = 5/6.

Step 2: Use the Point-Slope Equation

Now that we have the slope, we can use one of the points and the slope to write the equation of the line in point-slope form: y - y1 = m*(x - x1). Using Point1(0,0) and the value for m found in step 1, we have: y - 0 = (5/6)*(x - 0). After simplifying we have y = (5/6)*x.

Key Concepts

Slope Calculation

To find the equation of a line, we first need the slope, a crucial component when writing it in point-slope form. The slope of a line connects two points on the line by indicating how steep the line is. It is essentially a number that tells us how fast the line rises or falls as we move along it horizontally.
The formula to calculate slope is simple and is represented as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here:

• \( (x_1, y_1) \) represents the coordinates of the first point.
• \( (x_2, y_2) \) represents the coordinates of the second point.

In practice, let's calculate the slope using the points \((0, 0)\) and \((-6, -5)\). Plugging these into the slope formula, we compute: \[ m = \frac{-5 - 0}{-6 - 0} = \frac{5}{6} \]. This tells us that for every 6 units we move to the right (positive x-direction), the line moves 5 units up (positive y-direction). A positive slope like this indicates the line goes upwards as you move from left to right.

Equation of a Line

With the slope calculated, we can proceed to determine the equation of the line in point-slope form, which is especially useful for finding a line's equation when you have one point and the slope. The point-slope form is expressed as:\[ y - y_1 = m(x - x_1) \].

• \( y_1 \) is the y-coordinate of the given point.
• \( x_1 \) is the x-coordinate of the given point.
• \( m \) is the slope, which we've calculated earlier.

Substituting the slope of \( \frac{5}{6} \) and the point \((0, 0)\), the equation becomes:\[ y - 0 = \frac{5}{6}(x - 0) \].Simplifying this, we find the equation of the line to be \( y = \frac{5}{6}x \). This linear equation tells us that the line crosses the origin and increases with a slope of \( \frac{5}{6} \).

Coordinate Geometry

Coordinate geometry allows us to geometrically interpret algebraic equations by converting them into points on a graph. It's like drawing a map of mathematics where numbers and equations are locations and routes. Understanding this is pivotal for solving problems involving lines, shapes, and other geometrical objects.
In coordinate geometry, every point on a plane is defined by two coordinates, usually in the form \((x, y)\). This notation helps us locate and describe various geometrical figures like lines or curves.
For any line equation, like the one we've just calculated, \( y = \frac{5}{6}x \), we can:

• Plot multiple points by substituting different values for \( x \) and calculating corresponding \( y \) values.
• Draw the line on a coordinate plane by connecting these points.
• Visualize the slope as the angle or steepness of the line connecting these points.

This visualization helps us understand not only how lines behave algebraically but also how they interact across geometric space. Coordinate geometry bridges the gap between abstract algebra and tangible graphs, enhancing our comprehension of spatial relationships.