Problem 40 - Elementary Algebra

Question

Determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. (Objective 2) $$5 x-2 y=0$$

Step-by-Step Solution

Answer

The slope is \( \frac{5}{2} \) and the y-intercept is 0. Graph a line through the origin with this slope.

Step 1: Equation in Slope-Intercept Form

To determine the slope and intercept, first convert the given equation into the slope-intercept form, which is given by \( y = mx + b \). Start by isolating \( y \) on one side of the equation:Given equation: \( 5x - 2y = 0 \). Subtract \( 5x \) from both sides:\[ -2y = -5x \]Now, divide each term by \(-2\) to solve for \( y \):\[ y = \frac{5}{2}x \].

Step 2: Identify the Slope and Y-Intercept

Now that the equation is in slope-intercept form \( y = mx + b \), we can easily identify the slope \( m \) and the y-intercept \( b \).For \( y = \frac{5}{2}x \):- Slope \( m = \frac{5}{2} \).- Y-intercept \( b = 0 \). This means the line crosses the y-axis at point \( (0, 0) \).

Step 3: Graph the Line

Use the slope and the y-intercept to graph the line. 1. Plot the y-intercept \( b = 0 \) at the origin \((0, 0)\) on the graph.2. Use the slope \( \frac{5}{2} \), which means that for every increase of 2 units along the x-axis, the y value increases by 5 units. From the origin, move 2 units to the right (positive direction) on the x-axis and 5 units up (positive direction) to plot the second point.3. Draw a straight line through these points, extending in both directions.

Key Concepts

Graphing Linear Equations

Graphing a linear equation involves plotting points on a graph to represent the equation visually. A linear equation, in its typical form, is represented by the equation: \( y = mx + b \). Here, \( m \) indicates the slope and \( b \) is the y-intercept.
The process starts with converting any given equation into this slope-intercept form. This allows you to clearly identify the slope and the y-intercept, which are crucial for graphing. Using the information provided by the slope and y-intercept, you can plot the line accurately.

• First, determine the y-intercept. This is where the line will cross the y-axis. Plot this point on the graph.
• Next, use the slope to determine the next point. The slope describes how steep the line is and its direction. For example, if the slope is \( \frac{5}{2} \), from the y-intercept, move 2 units along the x-axis and 5 units up the y-axis to find another point on the line.
• Finally, draw the line through the plotted points. Extend the line in both directions to complete the graph.

Remember, the key to graphing is accurately using the slope and intercept to establish the line's position on a coordinate plane.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It is usually represented by the letter \( m \) in the equation \( y = mx + b \). The slope indicates how much the y-value of a point on the line changes for a given change in the x-value.

• A positive slope means the line ascends from left to right; a negative slope means it descends.
• The greater the slope’s absolute value, the steeper the line.
• A slope of zero indicates a horizontal line, while an undefined slope indicates a vertical line.

For an equation like \( y = \frac{5}{2}x \), the slope is \( \frac{5}{2} \). This indicates that for every 2 units you move to the right along the x-axis, the y-value increases by 5 units. The concept of slope is fundamental, as it not only describes a line's direction but also how rapidly it changes along a graph.

Y-Intercept

The y-intercept is a critical feature of the line that shows where it crosses the y-axis. In any line equation written as \( y = mx + b \), \( b \) represents the y-intercept. This value indicates the point on the graph where the line intersects the y-axis, corresponding to the x-value of zero.
For example, in the equation \( y = \frac{5}{2}x \), the y-intercept is \( 0 \). This tells us that when \( x = 0 \), \( y = 0 \), which means the line goes through the origin \((0,0)\).
Understanding the y-intercept is essential when graphing a line since it serves as the starting point for graphing. The line will rise or fall from this point, depending on the slope.

• A line with a non-zero y-intercept \( b \) crosses the y-axis at that point.
• For a graph with a y-intercept of zero, if the line passes through the origin, any increase or decrease due to the slope will branch out from this central point.

Thus, mastering this concept ensures you draw accurate crossings and representations when plotting your linear equations.