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Chapter 4: Problem 24

Graph each equation using the slope and \(y\)-intercept. \(4 x-3 y=-6\)

Short Answer

Expert verified
Plot \((0, 2)\), then from there move 4 up and 3 right; draw the line through the points.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

Start with the given equation \(4x - 3y = -6\). We want to get it into the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. First, solve for \(y\):\[ 4x - 3y = -6 \]\[ -3y = -4x - 6 \]\[ y = \frac{4}{3}x + 2 \]. Now the equation is in slope-intercept form: \(y = \frac{4}{3}x + 2\).
02

Identify the Slope and y-Intercept

From the equation \(y = \frac{4}{3}x + 2\), we can see that the slope \(m\) is \(\frac{4}{3}\) and the y-intercept \(b\) is 2. This means the line crosses the y-axis at the point \((0, 2)\).
03

Plot the y-Intercept

On a graph, start by plotting the y-intercept. This is the point \( (0, 2) \) on the y-axis.
04

Use the Slope to Plot Another Point

From the y-intercept \((0, 2)\), use the slope \(\frac{4}{3}\) to plot another point. The slope \(\frac{4}{3}\) means you rise 4 units and run 3 units. Starting at \((0, 2)\), move up 4 units to 6, and right 3 units to 3, marking another point at \((3, 6)\).
05

Draw the Line

With both points, \((0, 2)\) and \((3, 6)\), plotted, draw a straight line through these points. Extend the line across the graph, and you will have the graph of the equation \(y = \frac{4}{3}x + 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a linear equation is a way to express straight lines using the formula \( y = mx + b \). It is a convenient way to quickly identify the slope and y-intercept of a line. Here:
  • \( y \) represents the dependent variable, typically found on the vertical axis of a graph.
  • \( x \) is the independent variable, usually placed on the horizontal axis.
  • \( m \) denotes the slope of the line, indicating how steep it is.
  • \( b \) is the y-intercept, which tells you where the line crosses the y-axis.
In our equation \( 4x - 3y = -6 \), we first rearrange to solve for \( y \) so we can easily discern both the slope and y-intercept. This involves isolating \( y \) on one side of the equation by performing algebraic operations to put it in the form \( y = \frac{4}{3}x + 2 \). Now the equation is ready to be graphed, instantly showing us how the line behaves.
Slope
The slope is a measure of the line's steepness and direction. It tells us how much \( y \) changes for a change in \( x \). In the slope-intercept form \( y = mx + b \), the slope is represented by \( m \).
  • If the slope is positive, as in our example where \( m = \frac{4}{3} \), the line rises as it moves from left to right.
  • If the slope were negative, the line would decline going in the same direction.
  • A larger slope value indicates a steeper line, while a smaller value means a gentler incline.
In the context of our graph, the slope \( \frac{4}{3} \) tells us to move up 4 units (rise) and 3 units to the right (run). This simple rise-over-run method makes it easy to plot additional points on the graph after starting from the y-intercept. By using the slope, we ensure the line accurately represents the equation.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. This is crucial as it gives a starting point for graphing the line. In our slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \).
  • In the equation \( y = \frac{4}{3}x + 2 \), the y-intercept \( b \) is 2.
  • This means the line will cross the y-axis at the point \( (0, 2) \).
  • Every graph of a line in slope-intercept form includes a clear y-intercept for quick plotting.
To begin graphing, you start by placing a point at \( (0, 2) \) on the graph. This point makes it simple to utilize the slope next, as you extend the line according to its incline. Understanding the y-intercept helps you set the baseline for any linear equation graphing task, ensuring you accurately represent the line's position on the plane.

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