Problem 14 - Elementary Algebra

Question

Graph the equations. $$ y=\frac{3}{5} x $$

Step-by-Step Solution

Answer

Answer: The slope of the linear equation is (3/5) and the y-intercept is 0.

Step 1: Identify the slope and y-intercept

The equation is written in the form of y = mx + b, where m represents the slope and b represents the y-intercept. In this case, the slope (m) is (3/5) and the y-intercept (b) is 0. # Step 2: Plot the y-intercept on the graph

Step 2: Plot the y-intercept

As the y-intercept (b) is 0, we know that the equation crosses the y-axis at the point (0, 0). Place a point on the graph at the origin (0, 0). # Step 3: Use the slope to plot a second point

Step 3: Use the slope to plot a second point

The slope (m) is (3/5), which means that for every increase of 5 units in the x-direction, there is an increase of 3 units in the y-direction. Starting at the y-intercept (0, 0), move 5 units to the right and 3 units up to find the second point (5, 3). Place a point on the graph at this location. # Step 4: Draw a straight line through the points

Step 4: Draw a straight line through the points

Connect the two points (0, 0) and (5, 3) with a straight line, which represents the graph of the linear equation y=(3/5)x.

Key Concepts

Slope

The concept of a slope is essential when graphing linear equations. The slope represents how steep a line is on the graph. In mathematical terms, it describes the 'rise over run,' which is the change in vertical distance divided by the change in horizontal distance.

If you imagine a straight water slide, the slope would be the angle at which the slide descends. For the equation \(y = \frac{3}{5}x\), the slope is \(\frac{3}{5}\). This ratio tells you that for every 5 steps horizontally to the right, the line ascends 3 steps vertically upward.

Slopes can be positive, negative, zero, or undefined:

• A positive slope means the line ascends from left to right.
• A negative slope means the line descends from left to right.
• A zero slope means the line is perfectly horizontal.
• An undefined slope means the line is vertical.

Understanding slopes helps in predicting and understanding the behavior of linear relationships on a coordinate plane.

Y-intercept

The y-intercept is where a line crosses the y-axis on a graph. In the equation of a line, \(y = mx + b\), the y-intercept is represented by \(b\). It shows the starting point of the line when \(x = 0\).

For the equation \(y = \frac{3}{5}x\), the y-intercept is 0. This tells us that the line passes through the origin, at the point (0, 0).

Here are some key points to remember about y-intercepts:

• The y-intercept is crucial for plotting the initial point of a line on the graph.
• It's the value of \(y\) when \(x = 0\).
• It acts as a starting point to plot the line using the slope.

Grasping the concept of y-intercepts helps establish the foundation of linear equations on a coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graphically represent equations. It's composed of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical), intersecting at the origin (0,0).

The coordinate plane allows for:

• Plotting points using pairs of numbers called coordinates (x, y).
• Visualizing the slope and y-intercept of linear equations.
• Observing the relationships and intersections between multiple lines.

Each point on the plane corresponds to a specific location dictated by its x (horizontal) and y (vertical) coordinates. When graphing linear equations like \(y = \frac{3}{5}x\), the coordinate plane is the stage where the equation comes to life, showcasing how the line behaves as \(x\) and \(y\) values change.

Linear Relationship

A linear relationship refers to a constant, straight-line association between two variables. In mathematical terms, it’s characterized by the equation \(y = mx + b\), where \(m\) and \(b\) represent constants.

Here's what defines a linear relationship:

• The relationship between variables is proportional and consistent.
• The graph of a linear equation results in a straight line.
• Each increase in the independent variable (x) results in a consistent change in the dependent variable (y).

In the equation \(y = \frac{3}{5}x\), the linear relationship can easily be visualized on a coordinate plane. The line’s unwavering path from the origin (0, 0) upwards through (5, 3) exemplifies this consistent pattern. Truly understanding linear relationships paves the way for deeper insights into algebraic concepts and real-world applications like interpreting data trends and predicting outcomes.