Problem 45 - Precalculus Mathematics for Calculus

Question

Find the slope and \(y\) -intercept of the line and draw its graph. $$x+y=3$$

Step-by-Step Solution

Answer

Slope is -1, y-intercept is 3.

Step 1: Rewrite in Slope-Intercept Form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To rewrite the equation \(x + y = 3\) in slope-intercept form, solve for \(y\). Start by subtracting \(x\) from both sides of the equation to isolate \(y\):\[ y = -x + 3 \]

Step 2: Identify the Slope and Y-Intercept

Now that the equation is in the form \(y = mx + b\), identify the slope \(m\) and the y-intercept \(b\). From the equation \(y = -x + 3\), the slope \(m\) is \(-1\) and the y-intercept \(b\) is \(3\).

Step 3: Draw the Graph Using Slope and Y-Intercept

To graph the line, start by plotting the y-intercept on the coordinate plane. The y-intercept \(b = 3\) means the line crosses the y-axis at \((0, 3)\). From this point, use the slope \(m = -1\) to find another point on the line. A slope of \(-1\) means that for every 1 unit you move right, move 1 unit down. Plot a second point at \((1, 2)\). Draw a straight line through these points, extending it in both directions.

Key Concepts

Slope-Intercept Form

When working with linear equations, the slope-intercept form is a highly useful way to express the equation of a line. The form is written as \(y = mx + b\), where:

• \(m\) is the slope of the line
• \(b\) is the y-intercept

The primary advantage of using this form is its simplicity in identifying the slope and y-intercept directly from the equation. For instance, when you have an equation like \(y = -x + 3\), it's immediately clear that the slope \(m\) is \(-1\) and the y-intercept \(b\) is \(3\).
To convert any linear equation into the slope-intercept form, solve for \(y\) to isolate it on one side of the equation. Taking the original equation \(x + y = 3\) as an example, you subtract \(x\) from both sides to get \(y = -x + 3\), putting the equation neatly into the form \(y = mx + b\).

Slope

The slope of a line is a measure of its steepness and direction. Mathematically, the slope \(m\) is defined as "rise over run," or the change in \(y\) over the change in \(x\). In simpler terms, the slope tells us how much \(y\) changes for a one-unit change in \(x\).

• A positive slope means the line is rising as it moves from left to right.
• A negative slope means downwards from left to right.
• A zero slope means the line is horizontal.
• An undefined slope (division by zero) suggests a vertical line.

The slope is derived directly from the \(m\) in the equation \(y = mx + b\). For the line described by \(y = -x + 3\), the slope \(m\) is \(-1\). This indicates that for every step you take to the right along the \(x\)-axis, you will move one step down along the \(y\)-axis, signifying a downward slope.

Y-intercept

The y-intercept of a line is the point where it crosses the y-axis. This specifically occurs when \(x = 0\) since the y-intercept shows where the line intersects the vertical axis. In terms of the slope-intercept form \(y = mx + b\), the y-intercept is denoted by \(b\).
When graphing, the y-intercept is a crucial starting point. You can easily plot it on the graph, allowing you to draw the rest of the line using the slope. For the equation \(y = -x + 3\), the y-intercept is \(3\). This means the line passes through the point \((0, 3)\), where it crosses the y-axis.
By plotting the y-intercept and using the slope to find another point, such as \((1, 2)\) given \(m = -1\), one can draw a straight line through these points to represent the equation on a graph.