Problem 76 - Algebra 1: Concepts and Skills

Question

Find the slope of the line that passes through the points. \((0,0)\) and \((4,5)\)

Step-by-Step Solution

Answer

The slope of the line that passes through the points (0,0) and (4,5) is 5/4.

Step 1: Identify the coordinates

First, identify the coordinates for (x1, y1) and (x2, y2). Here, (x1, y1) can be (0,0) and (x2, y2) can be (4,5).

Step 2: Plug into the slope formula

Next, plug the coordinates into the slope formula. The slope formula is 'slope = (y2 - y1) / (x2 - x1)'. So the equation will be (5-0) / (4-0).

Step 3: Simplify the equation

Simplify the equation to find the slope of the line. This simplifies to 5/4.

Key Concepts

Slope Formula

Understanding the slope formula is crucial when studying coordinate geometry. Essentially, the slope is a measure of how steep a line is. In algebra, the slope is often represented as 'm' and is calculated by the difference in the y-coordinates divided by the difference in the x-coordinates between two points on the line.

The standard formula is \[ m = \dfrac{y_2 - y_1}{x_2 - x_1} \]
When applying this formula, ensure you consistently label your points so (x1, y1) and (x2, y2) are from the same point. In the given problem, by placing point (0,0) as (x1, y1) and point (4,5) as (x2, y2), we calculate the slope as 5/4. This result represents the rate of change along the line for each unit moved horizontally.

Slope-Intercept Form

Once you understand the slope, you can delve into the slope-intercept form of a linear equation, which is another fundamental concept in algebra. This form is expressed as \[ y = mx + b \]
where 'm' represents the slope of the line, and 'b' is the y-intercept, the point where the line crosses the y-axis. This equation allows you to graph a line on a coordinate plane easily when you're given the slope and the y-intercept.

Using the slope found from our example (5/4), if we know the y-intercept, we can write the equation for the line. For example, if the y-intercept is 0 (as our first point suggests), the equation of the line through points (0,0) and (4,5) would be \[ y = \frac{5}{4}x \].

Algebraic Concepts

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve problems. The slope formula and slope-intercept form are both algebraic concepts used to solve various problems involving linear equations and functions.

These tools allow us to understand the relationships between quantities and how they change relative to each other. Whether finding the slope or writing an equation, algebra teaches us to work with unknowns and variables systematically, offering a way to model real-life situations into mathematical expressions.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, brings together algebra and geometry to solve problems using a coordinate plane. This field focuses on the study of geometric shapes like lines, segments, and curves, where each point can be identified by a pair of numerical coordinates.

In coordinate geometry, finding the slope is a fundamental task as it defines the direction and steepness of lines. By plotting points and using the slope, as we did with points (0,0) and (4,5), we can visualize and analyze the properties of geometric shapes. It is a powerful tool that links algebraic equations to graphical representations.