Question

Give the equation and graph for a line with y-intercept and slope given in Exercises. $$y \text { -intercept }=3 ; \text { slope }=-1$$

Short answer

Answer: The equation of the line is \(y = -x + 3\). The graph of the line starts at the y-intercept point (0, 3) and has a negative slope, meaning it moves downward to the right, creating a diagonal line.

Step-by-step solution

Identify slope and y-intercept values

We are given the y-intercept \((b = 3)\) and the slope \((m = -1)\).

Plug slope and y-intercept values into the slope-intercept form

We need to plug the values of \(m\) and \(b\) into the slope-intercept equation: $$y = mx + b$$ In this case, \(m = -1\) and \(b = 3\), so our equation will be: $$y = -1x + 3$$

Simplify the equation (if needed)

In this case, our equation is already simplified: $$y = -x + 3$$ This is our final equation.

Graph the line

To graph the line, we will use the y-intercept and the slope: 1. Plot the y-intercept point \((0, 3)\) on the coordinate plane. 2. Since the slope is \(-1\), this means that the line goes down one unit and moves to the right one unit for every point. 3. Continue plotting points using the slope from the y-intercept, creating a line that goes through all the points. 4. Once you've plotted several points, draw a straight line through them, and extend the line in both directions. 5. Label the line with its equation: \(y = -x + 3\). Now we have the equation and graph of the line with the given y-intercept and slope.

Key concepts

Y-Intercept

Understanding the y-intercept of a line is crucial when dealing with linear equations. The term y-intercept refers to the point where the line crosses the y-axis on a graph. It is represented by the coordinate (0, b), where b is the y-intercept value. In our exercise, the y-intercept given is 3, which means the line crosses the y-axis at the point (0, 3). This point is essential for graphing the line, as it serves as a starting point from which we can use the slope to find other points.

Whenever you look at the equation of a line in the slope-intercept form, (y = mx + b), the y-intercept is always the constant b. This value tells us not only where to start plotting our graph but also provides insight into the behavior of the line. For example, a positive y-intercept means the line crosses the y-axis above the origin, while a negative one means the line crosses below the origin.

Slope

The slope of a line is a measure of its steepness or incline and is usually denoted as m in the slope-intercept form equation, y = mx + b. The slope indicates how much the y value of a point on the line changes for a one-unit change in the x value. In other words, it shows the rate at which y increases or decreases as x increases. A positive slope means the line rises from left to right, while a negative slope means the line falls.

In the exercise, the slope given is -1, indicating that for each step to the right along the x-axis, the line will go down by one unit. It's a straightforward way to continue plotting points starting from the y-intercept and forming a line. Slope is a fundamental concept as it defines the direction and angle of the line on a graph.

Linear Equations

Linear equations form the basis for much of algebra and are equations that produce straight lines when graphed on a coordinate plane. Their general form in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. These equations are incredibly useful because they can model real-world phenomena such as speed, cost, and other rates of change.

One of the key characteristics of linear equations is their predictability. They allow us to calculate unknowns based on known values. The slope-intercept form of a linear equation gives the most direct insight into a line's slope and where it crosses the y-axis, thus assisting in predicting and understanding behavior simply by looking at the equation's coefficients.

Graphing Linear Equations

Graphing linear equations involves creating a visual representation of the equation on the xy-plane. The easiest way to graph a line is using the slope-intercept form of its equation. The algorithm is simple: start by plotting the y-intercept, then use the slope to find additional points. After plotting a few points, draw a line through these points that extends indefinitely in both directions, since linear equations represent lines that continue without end.

In the exercise example, after plotting the y-intercept (0, 3), we use the slope, -1, to plot other points by moving horizontally from the y-intercept and applying the 'rise over run' method — down one unit and right one unit, in this case. Finally, a straight edge helps ensure that the line is straight and accurate. Graphing not only provides a visual summary of the equation but also aids in understanding the relationship between variables.