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Chapter 3: Problem 19

Use the slope-intercept form to graph each equation. See Examples 2 and 3. $$ 4 x-7 y=-14 $$

Short Answer

Expert verified
Graph the equation by plotting the y-intercept at (0, 2) and using the slope \( \frac{4}{7} \) to plot another point at (7, 6), then draw the line through these points.

Step by step solution

01

Rewrite equation in slope-intercept form

Start by rearranging the given equation into the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.Our equation starts as: \[ 4x - 7y = -14 \] Subtract \( 4x \) from both sides:\[ -7y = -4x - 14 \] Now, divide every term by \( -7 \):\[ y = \frac{4}{7}x + 2 \] This is the slope-intercept form, where the slope \( m = \frac{4}{7} \) and the y-intercept \( b = 2 \).
02

Plot the y-intercept

To begin graphing, locate the y-intercept on the graph. The y-intercept \( b \) is where the line crosses the y-axis. From the equation \( y = \frac{4}{7}x + 2 \), the y-intercept is 2.Put a point at (0, 2) on the graph.
03

Use the slope to find another point

The slope \( m = \frac{4}{7} \) means that for every increase of 4 units in the y-direction, the graph increases 7 units in the x-direction. From the y-intercept (0, 2), move up 4 units to reach y = 6, and right 7 units to reach x = 7, giving you the point (7, 6).Plot this second point on the graph at (7, 6).
04

Draw the line

With the two points (0, 2) and (7, 6) plotted, draw a straight line through them. Extend this line across the graph, ensuring it is straight and extends in both directions.

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Key Concepts

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

Graphing Equations
Graphing equations, especially linear ones, can be a straightforward task once you understand the basic concepts. It involves plotting points on a graph that represent the equation. In the slope-intercept form, each equation represents a line.
To graph an equation:
  • First, convert it into the slope-intercept form (\( y = mx + b \)).
  • Next, locate the y-intercept on the graph. This is the starting point where your line will cross the y-axis.
  • Then, use the slope to find another point. The slope tells you how to move from the y-intercept to find more points along the line.
  • Finally, draw a straight line through these points.
Understanding these steps will make it easier to visualize and accurately graph any linear equation.
Linear Equations
Linear equations are mathematical expressions where the highest power of the variable is one. These equations graph as straight lines on a two-dimensional plane. They are commonly written in the form \( ax + by = c \).
When dealing with linear equations:
  • Recognize that they have constant rates of change, making their graphs predictable.
  • Converting them to slope-intercept form \( (y = mx + b) \) simplifies understanding.
  • They are easy to solve or graph, since finding the y-intercept and slope gives you necessary points for graphing.
Linear equations appear in various real-world situations, helping describe relationships as constant, straight lines.
Slope and Y-Intercept
The slope and y-intercept are key components of the slope-intercept form of a linear equation \( (y = mx + b) \).
  • Slope (m): This indicates the steepness of a line. A slope of \( rac{4}{7} \), for example, means for every 7 units moved horizontally (x-direction), you move 4 units vertically (y-direction).
  • Y-Intercept (b): This is where the line crosses the y-axis. It represents the output value when the input (x) is zero. For example, if \( b = 2 \), the line crosses the y-axis at (0, 2).
Understanding these components helps with graphing and interpreting linear relationships, as these values determine the line鈥檚 path on a graph.

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Most popular questions from this chapter

Solve each equation for \(y.\) \(y-(-6)=2(x-4)\)

Solve. See the Concept Check in this section. Is the graph of \((3,0)\) in the same location as the graph of \((0,3) ?\) Explain why or why not.

Find the value of \(x^{2}-3 x+1\) for each given value of \(x .\) See Section 1.7. $$ -1 $$

Determine whether each pair of lines is parallel, perpendicular, or neither. \(6+4 x=3 y\) \(3 x+4 y=8\)

Find the slope of each line. \(y=-3\)
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