Problem 44 - Algebra and Trigonometry

Question

Find the slope and \(y\)-intercept of the line and draw its graph. \(2 x-5 y=0\)

Step-by-Step Solution

Answer

Slope is \(\frac{2}{5}\), y-intercept is \(0\).

Step 1: Rearrange Equation to Slope-Intercept Form

Start with the given equation: \[2x - 5y = 0\].The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Rearrange the equation to this form by solving for \(y\). First, isolate \(5y\) by moving \(2x\) to the other side:\[-5y = -2x\].Next, divide every term by \(-5\) to solve for \(y\):\[y = \frac{2}{5}x\].

Step 2: Identify Slope and Y-Intercept

From the rearranged equation \(y = \frac{2}{5}x\), identify the slope and y-intercept:- The slope \(m\) is \(\frac{2}{5}\).- The y-intercept \(b\) is \(0\) because there is no constant term added to \(\frac{2}{5}x\). Therefore, the y-intercept is at point \((0,0)\).

Step 3: Plot the Graph

To draw the graph, use the y-intercept and another point derived from the slope:1. Plot the y-intercept \((0, 0)\) on the graph.2. Use the slope \(\frac{2}{5}\) which indicates a 'rise' of 2 units for every 5 units of 'run'. From the y-intercept, move up 2 units and right 5 units to find another point \((5, 2)\).3. Draw a straight line through the points \((0,0)\) and \((5,2)\). This line represents the equation \(2x - 5y = 0\).

Key Concepts

Slope

The slope of a line is a measure of its steepness. It's basically telling you how "slanted" the line is. Imagine climbing a hill: a steep hill has a high slope, while a gentle hill has a low slope.
In the context of linear equations, the slope is often represented by the letter "m" in the slope-intercept form equation, which looks like this:

• The formula: \[ y = mx + b \]

Now, how do you find the slope?
For any line, you can find the slope using the formula:

• \[ m = \frac{{\text{{change in }} y}}{{\text{{change in }} x}} = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]

In our given equation, after converting it to slope-intercept form \( y = \frac{2}{5}x \), it's clear that the slope \( m \) is \( \frac{2}{5} \). Another way of looking at slope, is thinking about how much you go up or down (rise) for every step you take sideways (run).
For instance, with a slope of \( \frac{2}{5} \), this means for every 5 units you go horizontally (to the right if positive), you go up 2 units.
This helps in visualizing how the line moves across the graph.

y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. This is a crucial part of understanding where the line "begins" on the graph. Think of it as the spot where your path intersects the main vertical street.
In the slope-intercept formula, the y-intercept is represented by \( b \). That makes the equation:

• \[ y = mx + b \]

Finding the y-intercept is simple when your equation is in the slope-intercept form! It's just the value of \( b \).
For the equation derived from our problem, \( y = \frac{2}{5}x \), the y-intercept \( b \) is 0. This means the line crosses the y-axis at the origin point \( (0, 0) \).
This provides a starting point which you can use to draw or analyze the graph of the line.

Slope-Intercept Form

The slope-intercept form of a line is a very handy way to write the equation of a line. When your line's equation is in this form, it's easy to see both its slope and y-intercept right away. How cool is that?!!
The formula is:

• \[ y = mx + b \]

Here, \( m \) is the slope and \( b \) is the y-intercept. This format gives you direct insight into how the line behaves on a graph:

• The slope \( m \) tells you how the line angles upward or downward.
• The y-intercept \( b \) tells you where the line lands on the y-axis.

Converting from a general equation to the slope-intercept form is often just a matter of solving for \( y \). In our exercise, we converted \( 2x - 5y = 0 \) to \( y = \frac{2}{5}x \). This clearly reveals \( m = \frac{2}{5} \) and \( b = 0 \).
This form not only simplifies finding key features of a line but also makes graphing it straightforward by starting at the y-intercept and using the slope.