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Chapter 3: Problem 69

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-3 y=27\)

Short Answer

Expert verified
y-intercept: (0, -9). x-intercept: (27, 0). Slope: \(\frac{1}{3}\).

Step by step solution

01

- Rearrange the Equation for Intercepts

To find the intercepts and slope, rearrange the equation into the slope-intercept form: \[ x - 3y = 27 \]into \[ y = mx + b \].
02

- Find the y-intercept

Substitute \(x = 0\) into the rearranged equation and solve for \(y\): \[ 0 - 3y = 27 \] \[ -3y = 27 \] \[ y = -9 \]. The \(y\)-intercept is \((0, -9)\).
03

- Find the x-intercept

Substitute \(y = 0\) into the original equation and solve for \(x\): \[ x - 3(0) = 27 \] \[ x = 27 \]. The \(x\)-intercept is \((27, 0)\).
04

- Rearrange to slope-intercept form

Rearrange the original equation to the form \(y = mx + b\). Start with \[ x - 3y = 27 \] Subtract \(x\) from both sides: \[ -3y = -x + 27 \] Divide by \(-3\): \[ y = \frac{1}{3}x - 9 \].
05

- Identify the slope

The slope \(m\) in the equation \(y = mx + b\) is the coefficient of \(x\). Therefore, the slope is \[ m = \frac{1}{3} \].

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Key Concepts

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

y-intercept
The y-intercept of a line is where the line crosses the y-axis. At this point, the value of x is always zero. To find the y-intercept, we substitute x = 0 into the given equation and solve for y. In the exercise, we start with the equation:
\[ x - 3y = 27 \]
Substituting x = 0, we get:
\[ 0 - 3y = 27 \]
Solving for y, we divide both sides by -3:
\[ y = -9 \]
Hence, the y-intercept is the point (0, -9). This point tells us where the line meets the y-axis. It is crucial because it provides a starting point for graphing the line.
x-intercept
The x-intercept of a line is where the line crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept, we substitute y = 0 into the given equation and solve for x. From the equation:
\[ x - 3y = 27 \]
Substituting y = 0, we get:
\[ x - 3(0) = 27 \]
This simplifies to:
\[ x = 27 \]
Thus, the x-intercept is the point (27, 0). This point indicates where the line intersects the x-axis, which is another key information for graphing the line.
slope formula
The slope of a line measures its steepness and is represented by the letter 'm'. The slope formula is given by:
\[ m = \frac{ (y_2 - y_1) }{ (x_2 - x_1) } \]
However, in the given exercise, to find the slope more easily, we first rearrange the equation to the slope-intercept form. Starting from:
\[ x - 3y = 27 \]
Subtract x from both sides:
\[ -3y = -x + 27 \]
Then, divide by -3:
\[ y = \frac{1}{3}x - 9 \]
In the rearranged equation \[ y = mx + b \], the slope 'm' is identified as the coefficient of x. Thus, here the slope is \[ m = \frac{1}{3} \]. Knowing the slope is vital for understanding how the line rises or falls as it moves from left to right.
slope-intercept form
The slope-intercept form of a line's equation is given by \[ y = mx + b \]. In this form, 'm' represents the slope, and 'b' represents the y-intercept. Converting given equations into this form makes it easy to identify both the slope and y-intercept. For the exercise, starting from:
\[ x - 3y = 27 \]
Subtract x:
\[ -3y = -x + 27 \]
Divide by -3:
\[ y = \frac{1}{3}x - 9 \]
Here, it is clear that:
  • The slope (m) is \[ \frac{1}{3} \]
  • The y-intercept (b) is -9
The slope-intercept form provides a straightforward way to graph a line, as you can start at the y-intercept and use the slope to find other points on the line.

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