Problem 50 - Algebra 1

Question

Find the slope and the \(y\) -intercept of the graph of the equation. Then graph the equation. $$ 3 x-y=-5 $$

Step-by-Step Solution

Answer

The slope of the graph of the equation is 3 and the y-intercept is 5.

Step 1: Rewrite to Slope-Intercept Form

Rearrange the given equation \(3x - y = -5\) to the form \(y = mx + c\) by isolating \(y\). So, \(y = 3x +5\).

Step 2: Identify the Slope and y-Intercept

In the equation \(y = 3x + 5\), the slope \(m\) is 3 and the \(y\)-intercept \(c\) is 5.

Step 3: Graph the Equation

Plot the y-intercept (0,5) on the graph. Using the slope, move up 3 units and right 1 unit from the y-intercept to plot another point, then draw the line through these points. You can extend the line in both directions.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when you want to graph linear equations quickly and efficiently. It's the equation of a line in the form \( y = mx + b \), where \( m \) and \( b \) represent specific characteristics of the line: \( m \) is the slope, and \( b \) is the y-intercept. A line in this form makes it easy to identify these characteristics at a glance.

The beauty of slope-intercept form lies in its simplicity; it provides a clear path to follow when you're plotting a line on a graph. After identifying \( m \) and \( b \), you can immediately locate the y-intercept on the y-axis and then use the slope to find another point on the line. With two points established, you've got everything you need to draw the entire line. This is particularly helpful for students who are just beginning to understand how to graph equations, as it breaks down the process into manageable steps.

In the example provided, we transformed the equation \( 3x - y = -5 \) into the slope-intercept form \( y = 3x + 5 \). The rearrangement made it easier to identify the slope and y-intercept, thus simplifying the graphing process.

Slope of a Line

The slope of a line, represented by \( m \) in the slope-intercept form, is a measure of its steepness, or its 'tilt.' It's defined as the ratio of the rise (vertical change) over the run (horizontal change) between any two points on the line. A positive slope means that the line is inclining upwards as it moves from left to right, while a negative slope indicates that the line is declining.

When learning about the slope, there are a few key points to understand:

• The slope is constant for a linear equation, meaning it doesn't change no matter which points you choose on the line to calculate it.
• In a graph, each point is determined by an ordered pair \( (x, y) \), and the difference in the y-values over the difference in the x-values between two points gives us the slope.
• If the slope is a fraction, you can still plot points on the graph by using the numerator and denominator as the rise and run respectively.

With these fundamentals, you can interpret and graph any linear equation with confidence. For example, with the slope of 3 from the equation \( y = 3x + 5 \), you would rise 3 units and run 1 unit to arrive at the next point to draw your line.

Y-Intercept

The y-intercept is another key concept in graphing linear equations. It's the point where the line crosses the y-axis, and is represented as \( b \) in the slope-intercept form \( y = mx + b \). This single value tells you exactly where to begin plotting your line. It's particularly handy because you're guaranteed that one point on your line is at the coordinates \( (0, b) \) - this is because when \( x = 0 \), only the y-intercept is left to define the line's position along the y-axis.

Remembering the y-intercept is fundamental because:

• It's a fixed point and serves as an initial reference for plotting the linear equation.
• Understanding where the line crosses the y-axis helps in visualizing the graph even before it is drawn.
• It can provide insights into the context of a problem, such as a starting value or initial condition in real-world scenarios.

In our exercise, the y-intercept is 5, so you can put a dot at \( (0, 5) \) on your graph. From this point, the slope guides you to create the shape of the line as you plot it across the graph's plane.