Problem 53 - Precalculus Mathematics for Calculus

Question

Find the slope and \(y\) -intercept of the line and draw its graph. $$3 x-4 y=12$$

Step-by-Step Solution

Answer

The slope is \(\frac{3}{4}\), and the \(y\)-intercept is \(-3\).

Step 1: Identify the Equation Format

The given equation is in standard form: \(3x - 4y = 12\). To find the slope and intercept, we'll convert this to the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept.

Step 2: Solve for \(y\)

To convert the equation to slope-intercept form, solve for \(y\):\[3x - 4y = 12\]Subtract \(3x\) from both sides:\[-4y = -3x + 12\]Now, divide every term by \(-4\) to isolate \(y\):\[y = \frac{3}{4}x - 3\]

Step 3: Identify the Slope and \(y\)-Intercept

From the equation \(y = \frac{3}{4}x - 3\), we can identify that the slope \(m\) is \(\frac{3}{4}\) and the \(y\)-intercept \(b\) is \(-3\).

Step 4: Graph the Equation

To draw the graph, start by plotting the \(y\)-intercept (0, -3) on the graph. This is the point where the line crosses the \(y\)-axis. Next, use the slope \(\frac{3}{4}\), which means you go up 3 units and 4 units to the right, to plot another point. Connect these points with a straight line to complete the graph.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a common way to express the equation of a line because it shows the slope and the y-intercept clearly. It's written as \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis (where \( x = 0 \)).

To convert an equation from standard form to slope-intercept form, we solve for \( y \) in terms of \( x \). This involves rearranging the terms to isolate \( y \) on one side and expressing the rest in terms of \( x \). In the example, from the equation \( 3x - 4y = 12 \), we rearranged it to \( y = \frac{3}{4}x - 3 \). Here, the slope \( m \) is \( \frac{3}{4} \), and the y-intercept \( b \) is \(-3\). This form helps in easily reading off these important characteristics of the line.

Graphing Lines

Graphing lines involves visualizing a line on a Cartesian plane using information such as the slope and y-intercept from the line equation. To graph the equation \( y = \frac{3}{4}x - 3 \):

• Start by marking the y-intercept on the graph. Here, the intercept is -3, so you mark the point (0, -3) on the y-axis.
• Next, use the slope to locate another point. The slope \( \frac{3}{4} \) means from the intercepted point, rise 3 units up and run 4 units to the right. This leads to another point which can be (4, 0) based on these directions.
• Draw a straight line through these points. Extend the line on both sides beyond the points marked.

This visual form allows you to immediately see the angle of the slope and how steep the line is, as well as where it will cross the y-axis.

Standard Form of a Line

The standard form of a line equation is \( Ax + By = C \), where:

• \( A \), \( B \), and \( C \) are integers.
• \( A \) and \( B \) are not both zero.

This format is particularly useful for certain computational tasks, like finding intercepts or using elimination methods in systems of equations. Unlike the slope-intercept form, the standard form doesn't directly show the slope and y-intercept. However, they can be derived by transforming it into the slope-intercept form. In our example, the equation \( 3x - 4y = 12 \) is in standard form. By solving for \( y \), we converted it to \( y = \frac{3}{4}x - 3 \), finding both the slope (\( \frac{3}{4} \)) and y-intercept (\(-3\)), thus making it more convenient for graphing.