Problem 40 - Elementary Algebra

Question

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ 5 x-2 y=0 $$

Step-by-Step Solution

Answer

The slope is \(\frac{5}{2}\) and the y-intercept is 0. The line passes through (0,0) and (2,5).

Step 1: Identify the standard form

The given equation is \(5x - 2y = 0\). This is in the standard form of a linear equation, \(Ax + By = C\). Here, \(A = 5\), \(B = -2\), and \(C = 0\).

Step 2: Convert to slope-intercept form

To find the slope and y-intercept, convert the equation to the slope-intercept form, \(y = mx + b\). Start by isolating \(y\) in the equation: \(5x - 2y = 0\).

Step 3: Solve for y

Rearrange the equation \(5x - 2y = 0\) to solve for \(y\):\[\begin{align*} 5x - 2y &= 0 \ -2y &= -5x \ y &= \frac{5}{2}x \end{align*}\]Now the equation is in the form \(y = \frac{5}{2}x + 0\).

Step 4: Identify slope and y-intercept

From the equation \(y = \frac{5}{2}x + 0\), we identify the slope \(m = \frac{5}{2}\) and the y-intercept \(b = 0\).

Step 5: Graph the line

To graph the line: Start at the y-intercept (0,0). From this point, use the slope \(\frac{5}{2}\) to find another point. The slope tells us to rise 5 units and run 2 units. Plot the point (2,5). Connect these points with a straight line.

Key Concepts

Linear Equations

Linear equations are the foundation of algebra and are essential for understanding many mathematical concepts. They represent straight lines on a graph and can be expressed in various forms. A linear equation typically has two variables, usually denoted as \(x\) and \(y\), and is written in the standard form \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are constants. For instance, the equation \(5x - 2y = 0\) is a linear equation in standard form, where \(A = 5\), \(B = -2\), and \(C = 0\).

Linear equations are powerful because:

• They can model real-world situations where relationships are linear.
• They allow us to predict values by understanding the relationship between two variables.

Converting from standard to slope-intercept form helps in easily identifying the slope and y-intercept, which leads us to our next concept.

Graphing Lines

Graphing lines is a practical way to visualize linear equations and understand the relationship between \(x\) and \(y\). When you graph a linear equation like \(y = \frac{5}{2}x + 0\), you plot points on a coordinate plane and connect them to form a straight line.

Here's how to simplify the process of graphing:

• Identify the y-intercept, which is the point where the line crosses the y-axis. For our equation, it is (0,0).
• Use the slope to determine the direction and steepness of the line. A slope of \(\frac{5}{2}\) means that for every 2 units you move horizontally, you move 5 units vertically.
• Start at the y-intercept and apply the slope to find another point, such as (2,5).
• Draw a line through the plotted points to represent the equation visually.

This graphical representation helps us interpret the data and relationships intuitively, providing a clear view of how changes in \(x\) affect \(y\).

Slope-Intercept Form

The slope-intercept form of a linear equation is incredibly useful for quickly identifying key characteristics of a line. The general form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Understanding the components in \(y = nx + b\):

• The slope \(m\) indicates the line's steepness and direction. A positive slope like \(\frac{5}{2}\) means the line rises as it moves from left to right.
• The y-intercept \(b\) informs where the line crosses the y-axis. In our example, it's 0, implying the line passes through the origin.

The slope-intercept form simplifies graphing and analyzing linear equations, making it a handy tool for students tackling problems like the one given.