Problem 58 - Precalculus with Calculus Previews

Question

Graph the plane whose equation is given. $$ 3 x-y-6=0 $$

Step-by-Step Solution

Answer

Plot points (0,-6) and (1,-3) from the equation \( y = 3x - 6 \) and draw the line.

Step 1: Rewrite the Plane Equation in Slope-Intercept Form

To graph the plane, we first rewrite the given equation in slope-intercept form, which is \( y = mx + b \). Start by isolating \( y \) on one side of the equation.\[3x - y - 6 = 0\]Add \( y \) and 6 to both sides:\[-y = -3x + 6\] Finally, multiply every term by -1 to solve for \( y \):\[ y = 3x - 6 \] This form tells us that the plane intersects the y-axis at \( -6 \) and has a slope \( m = 3 \).

Step 2: Determine the Y-Intercept

The y-intercept is the point where the plane crosses the y-axis. From the equation \( y = 3x - 6 \), we see that the intercept is at \( (0, -6) \).This is the point where we will begin plotting the line on the graph.

Step 3: Use the Slope to Find Another Point

The slope of the line is \( 3 \), which means that for every unit increase in \( x \), \( y \) increases by 3 units. Starting from the y-intercept at \( (0, -6) \), move 1 unit to the right and 3 units up to find another point on the graph.This gives us the point \( (1, -3) \).

Step 4: Plot the Points and Draw the Line

Plot the points \( (0, -6) \) and \( (1, -3) \) on the graph. Once these points are plotted, draw a straight line through them, extending it in both directions, which represents the graph of the plane \( 3x - y - 6 = 0 \).

Step 5: Verify by Checking a Point on the Plane

To ensure accuracy, substitute a point from the line (other than the intercepts) back into the original equation to check if it satisfies the equation.For example, using \( (2, 0) \):\[3(2) - 0 - 6 = 0\]This simplifies to \(6 - 6 = 0\), which holds true, confirming the accuracy of our graph.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to quickly identify the slope and the y-intercept of a line. This form is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.
This makes it easier to graph a line because it gives you all the information you need right from the equation.To convert an equation to this form, you isolate \( y \) on one side of the equation. Consider the equation from our exercise, \( 3x - y - 6 = 0 \):

• First, move the \( y \) and constant to one side: \( -y = -3x + 6 \).
• Next, multiply every term by \(-1\) to get \( y = 3x - 6 \).

Once in slope-intercept form, it's easy to identify the slope and y-intercept, which are crucial for graphing the equation.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It occurs when \( x = 0 \). From the slope-intercept form \( y = mx + b \), the \( b \) value represents this intercept.
In the exercise, the equation is \( y = 3x - 6 \) which means:

• When \( x = 0 \), \( y = -6 \).

This tells us that the y-intercept is \( (0, -6) \). This point is the starting point for graphing the line on a coordinate plane.
Simply locate \( -6 \) on the y-axis and place a point there. This is the position where the line will meet the axis.

Slope of a Line

The slope of a line is a measure of its steepness and direction. In the equation \( y = mx + b \), \( m \) is the slope. The slope tells you how much \( y \) changes for a unit change in \( x \).
A positive slope increases upwards as you move to the right, while a negative slope decreases. In the exercise, the equation is \( y = 3x - 6 \) with slope \( m = 3 \):

• This means for a change of 1 in \( x \), \( y \) increases by 3.

To graph with this slope, start at the y-intercept \( (0, -6) \). Move 1 unit right (increasing \( x \)) and 3 units up (increasing \( y \)).
This lands on another point, \( (1, -3) \). These outstretched steps form the line on the graph.

Checking Solutions

Verifying your graph can be done by plugging a specific point from the line back into the original equation to see if it satisfies that equation.
This step ensures that you haven't made any errors in plotting.In the exercise, after plotting points like \( (0, -6) \) and \( (1, -3) \), you might choose another point, such as \( (2, 0) \), to verify:

• Substitute \( x = 2 \), \( y = 0 \) into the original equation \( 3x - y - 6 = 0 \).
• You get \( 3(2) - 0 - 6 = 0 \) which simplifies to \( 6 - 6 = 0 \).

The equation holds true. Therefore, the graph accurately represents the equation. It’s a good practice to check several points to enhance confidence in your graph. By ensuring the plotted line satisfies the equation, mistakes can be caught and corrected easily.