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Chapter 5: Problem 28

Sketch the graph of the given equation. Label the intercepts. $$3 x-7 y=21$$

Short Answer

Expert verified
The x-intercept is \((7, 0)\) and the y-intercept is \((0, -3)\).

Step by step solution

01

- Identify the equation format

The given equation is in the standard linear form: \(3x - 7y = 21\).
02

- Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\).\[3x - 7(0) = 21\]\[3x = 21\]\[x = 7\]The x-intercept is \((7, 0)\).
03

- Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\).\[3(0) - 7y = 21\]\[-7y = 21\]\[y = -3\]The y-intercept is \((0, -3)\).
04

- Plot the intercepts on the graph

Plot the points \((7, 0)\) and \((0, -3)\) on the coordinate plane. These points are the intercepts.
05

- Draw the line

Draw a straight line through the points \((7, 0)\) and \((0, -3)\). This line represents the equation \(3x - 7y = 21\).
06

- Label the intercepts

Label the points \((7, 0)\) as the x-intercept and \((0, -3)\) as the y-intercept on the graph.

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Key Concepts

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

X-Intercept
To understand the x-intercept, picture the point where the line crosses the x-axis on the coordinate plane. In simpler terms, it's where the line meets the horizontal line (x-axis). At this point, the value of y is always zero.

Finding the X-Intercept
  • In any linear equation, set y = 0.
  • Solve for x.
Taking our example equation, \(3x - 7y = 21\), we set y to zero and solve:

\[ 3x - 7(0) = 21 \] \[ 3x = 21 \]
\[ x = 7 \]
So, the x-intercept is \( (7, 0) \). This is the point where the line touches the x-axis.

Remember, on the graph, this is noted as (7, 0).
Y-Intercept
The y-intercept is where the line crosses the y-axis. This is the vertical line on the coordinate plane. At this point, the value of x is always zero.

Finding the Y-Intercept
  • In any linear equation, set x = 0.
  • Solve for y.
For our example equation \(3x - 7y = 21\), we set x to zero and solve:

\[ 3(0) - 7y = 21 \] \[ -7y = 21 \]
\[ y = -3 \]
So, the y-intercept is \( (0, -3) \). This is the point where the line touches the y-axis.

On the graph, we mark this as (0, -3).
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can graph points, lines, and curves. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

Each point on the plane is defined by an ordered pair \( (x, y) \). The x value tells us how far to move horizontally, and the y value shows how far to move vertically.

Steps to Graph a Line
  • Identify any intercepts, such as the x-intercept and y-intercept.
  • Plot these intercepts on the coordinate plane.
  • Draw a straight line through these points.
By plotting points such as \( (7, 0) \) and \( (0, -3) \), we extend an infinite line that represents our linear equation.
Standard Form of a Linear Equation
The standard form of a linear equation is represented as
\[Ax + By = C \]
Here, A, B, and C are constants. This format is handy for quickly finding intercepts and graphing lines.

Converting to Slope-Intercept Form
Sometimes it鈥檚 easier to understand and work with the equation when it is in slope-intercept form ( \( y = mx + b \) ), where m represents the slope and b represents the y-intercept.
  • Our example equation \(3x - 7y = 21\) can be re-arranged as:
  • \[-7y = -3x + 21 \]
  • \[ y = \frac{3}{7}x - 3 \]
Both forms provide unique insights and are valuable tools for different mathematical tasks. While the standard form easily shows intercepts, the slope-intercept form reveals the rate of change and starting point.

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Most popular questions from this chapter

Solve for \(x: \frac{x}{3}-\frac{x-1}{4}=\frac{x+1}{2}\)

Solve each of the following problems algebraically. Be sure to label what the variable represents. The width of a rectangle is 4 less than \(\frac{2}{3}\) its length. If the perimeter of the rectangle is 3 times the length, find its dimensions.

Sets of values are given for variables having a linear relationship. In each case, write the slope-intercept form for the equation of the line corresponding to the given set of values and answer the accompanying question. $$\begin{array}{|l|c|c|c|} \hline x \text { (Number of people waiting in line) } & 6 & 8 & 9 \\ \hline y \text { (Number of minutes waiting time) } & 10 & 15 & 17.5 \\ \hline \end{array}$$ What would the waiting time be if 15 people are waiting in line?

A seamstress can hem a skirt in 10 minutes and make the cuffs on a pair of pants in 15 minutes. On a certain day she has allotted 300 minutes to doing these tasks. (a) Let \(s\) represent the number of skirts she hems and \(p\) represent the number of pairs of pants she cuffs. Write an equation that reflects the given situation. (b) Sketch the graph of this relationship. Be sure to label the coordinate axes clearly. (c) If she hems 18 skirts, use the equation you obtained in part (a) to find the number of pairs of pants she cuffs.

Write an equation of the line satisfying the given conditions. Passing through \((5,6)\) with slope 0
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