Question

Give the equation and graph for a line with \(y\) -intercept equal to 3 and slope equal to -1.

Short answer

The equation for the line is y = -x + 3. To graph the line, plot the y-intercept point (0, 3), and then from that point, move 1 unit down and 1 unit to the right to plot the point (1, 2). Then, draw a straight line passing through these two points (0, 3) and (1, 2).

Step-by-step solution

Using the slope-intercept form

We are given the slope (m) as -1 and the y-intercept (b) as 3. The slope-intercept form of a linear equation is given by: y = mx + b We just need to plug in the given values of m and b into this equation.

Substitute m and b values

Now, we have: y = (-1)x + 3 or y = -x + 3 This is the equation of the line.

Graphing the line

To graph the line, follow these steps: 1. Plot the y-intercept point (0, 3) on the graph. 2. Since the slope is -1, from the y-intercept point, move 1 unit down and 1 unit to the right. 3. Plot this new point (1, 2). 4. Draw a straight line passing through the two points (0, 3) and (1, 2). Now, we have the equation and graph for a line with a y-intercept equal to 3 and a slope equal to -1. The equation is y = -x + 3, and the graph is a straight line passing through the points (0, 3) and (1, 2).

Key concepts

Slope-Intercept Form

The slope-intercept form is one of the simplest ways to represent a linear equation. It is written as \( y = mx + b \). Here, \( m \) is the slope of the line, and \( b \) is the y-intercept. This form is widely used because it directly reveals two important characteristics of a line:

• The slope \( m \), which tells us how steep the line is and the direction it goes (upward if positive, downward if negative).
• The y-intercept \( b \), which is the point where the line crosses the y-axis.

Knowing the values of \( m \) and \( b \) allows you to quickly understand and graph the line without additional calculations.

Graphing Lines

Graphing lines using the slope-intercept form is straightforward. It involves a clear process that requires minimal effort once you understand these two components. Starting with the y-intercept visually anchors the line on the graph. From there, the slope tells you how to find another point on the line.

• First, plot the y-intercept \( (0, b) \) on the graph.
• Then, use the slope, which is the rise over the run, to find another point. For instance, a slope of -1 means from the y-intercept, go one unit down and one unit to the right. This yields another coordinate point.
• Finally, connect these points with a straight line, extending it across the graph. This line represents the equation visually.

Graphing lines is a visual representation that helps to better understand the relationship between \( x \) and \( y \). It shows how changes in \( x \) affect \( y \).

Y-Intercept

The y-intercept \( b \) is a vital part of a linear equation. It indicates where the line crosses the y-axis. This is the value of \( y \) when \( x \) is zero, literally the starting point in many graph plots.A positive y-intercept means your line crosses the y-axis above the origin, while a negative one means it crosses below. This intercept provides immediate insight into part of the line's position on the graph. Once plotted, the y-intercept serves as a reference point for determining the rest of the line. Knowing only this value allows you to start graphing, teamed up with the slope, to shape the entire line path. Recognizing this simple point can ease understanding complex equations and graph-construction to depict real-world scenarios.