Problem 29 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ (0,0),(3,2) $$

Step-by-Step Solution

Answer

Answer: The equation of the line in slope-intercept form is $$y = \frac{2}{3}x$$.

Step 1: Determine the slope (m) from the given points

To find the slope of the line given the two points (0,0) and (3,2), we'll use the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ So, m = (2 - 0) / (3 - 0).

Step 2: Calculate the slope

Using the values from the previous step, we have: m = 2 / 3 The slope of the line is 2/3.

Step 3: Write the equation in slope-intercept form

Now that we have the slope (m = 2/3) and the y-intercept (b = 0), we can write the equation of the line in slope-intercept form (y = mx + b): y = (2/3)x + 0 Since the y-intercept term (0) doesn't affect the equation, we can simplify it to: y = (2/3)x So, the equation of the line in slope-intercept form is: $$ y = \frac{2}{3}x $$

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra and geometry, representing straight lines on a graph. They help us understand the relationship between variables, often depicted as "x" and "y." The general form of a linear equation is given by:

• y = mx + b

where "m" is the slope and "b" is the y-intercept. These equations tell us how "y" changes as "x" changes.
Linear equations can model real-world situations where there is a constant rate of change. For example, they can represent financial relationships, predict trends, or describe motions. By understanding linear equations, students gain valuable skills in calculating and interpreting changes over time.

Slope Calculation

Calculating the slope is a crucial step in understanding linear equations. The slope tells us the steepness and direction of a line. It's a measure of how much "y" increases or decreases as "x" changes. We find the slope using the formula:

• m = \( \frac{y_2 - y_1}{x_2 - x_1} \)

This formula involves two points on the line. "\((x_1, y_1)\)" and "\((x_2, y_2)\)." In our exercise, these points are (0, 0) and (3, 2). Inserting these into the formula, we calculate:

• m = \( \frac{2-0}{3-0} = \frac{2}{3} \).

The slope \( \frac{2}{3} \) means for every 3 units moved along "x," "y" changes by 2 units. A positive slope indicates an upward trend. This calculation is essential for forming the equation of the line.

Coordinate Geometry

Coordinate geometry involves plotting points, lines, and shapes on an x-y plane. It provides a visual representation of algebraic equations. This branch of mathematics merges algebra with geometry, allowing calculations and graphing to explain spatial relationships.
In coordinate geometry, each point has coordinates like (x, y) that specify its exact location on the plane. For instance, the given points are (0,0) and (3,2). Plotting these shows how they align on the line described by our equation. This visual process makes it easier to understand how equations describe lines and other geometric figures.
The equation of a line, like \( y = \frac{2}{3}x \), tells us precisely how it behaves geometrically. It describes location, direction, and slope—all visible on the graph. Mastery of coordinate geometry is invaluable for solving diverse math problems and understanding complex systems.